There is an interesting review by Ron Solomon of a paper in this area, which has been featured on the Beyond Reviews blog. In particular, he outlines the broad tactics that people are using in CFSG II, and some of the content that will be going into volume 7.
Also, Inna Capdeboscq apparently gave an outline of volume 8, or at least a chunk of it, at the Asymptotic Group Theory conference in Budapest. This was mentioned by Peter Cameron on his blog, sadly with no detail! If anyone can get a whiff of what she said, I would be grateful.
EDIT 15 October 2016 I emailed the group-pub mailing list and was told second-hand that Ron Solomon 'has hopes' volume 7 will be submitted next year.
EDIT 27 March 2018 Thanks to Timothy Chow in a comment on another answer, here is the link to the published version of Volume 7. So now the countdown to Volume 8 starts...
EDIT 22 June 2018 Even better news: Volume 8
...is near completion and promised to the AMS by August 2018. The completion of Volume 8 will be a significant mathematical milestone in our work. (source)
Also (from the same article):
We anticipate that there will be twelve volumes in the complete series [GLS], which we hope to complete by 2023.
Considerable work has been done on this problem [the bicharacteristic case], originally by Gorenstein and Lyons, and more recently by Inna Capdeboscq, Lyons, and me. We anticipate that this will be the principal content of Volume 9 [GLS], coauthored with Capdeboscq.
When p is odd, there is a major 600-page manuscript by Gernot Stroth treating groups with a strongly p-embedded subgroup, which will appear in the [GLS] series, probably in Volume 11. There are also substantial drafts by Richard Foote, Gorenstein, and Lyons for a companion volume (Volume 10?), which together with Stroth’s volume will complete the p-Uniqueness Case.
It would be wonderful to complete our series by 2023, the sixtieth anniversary of the publication of the Odd Order Theorem. Given the state of Volumes 8, 9, 10, and 11, the achievement of this goal depends most heavily on the completion of the e(G) = 3 problem. It is a worthy goal.
EDIT Mar 2019 Volume 8 has been published. The page listing the available volumes, along with links to more details is here.
The summary of this volume is as follows:
This book completes a trilogy (Numbers 5, 7, and 8) of the series The Classification of the Finite Simple Groups treating the generic case of the classification of the finite simple groups. In conjunction with Numbers 4 and 6, it allows us to reach a major milestone in our series—the completion of the proof of the following theorem:
Theorem O: Let G be a finite simple group of odd type, all of whose proper simple sections are known simple groups. Then either G is an alternating group or G is a finite group of Lie type defined over a field of odd order or G is one of six sporadic simple groups.
Put another way, Theorem O asserts that any minimal counterexample to the classification of the finite simple groups must be of even type. The work of Aschbacher and Smith shows that a minimal counterexample is not of quasithin even type, while this volume shows that a minimal counterexample cannot be of generic even type, modulo the treatment of certain intermediate configurations of even type which will be ruled out in the next volume of our series.
EDIT February 2021 Volume 9 has now been published. From the preface:
This book contains a complete proof of Theorem $\mathcal{C}_5$, which covers the “bicharacteristic” subcase of the $e(G) \ge 4$ problem. The outcome of this theorem is that $G$ is isomorphic to one of the six sporadic groups for which $e(G)\ge 4$, or one of six groups of Lie type which exhibit both characteristic 2-like and characteristic 3-like properties. Finally, in Chapter 7, we begin the proof of Theorem $\mathcal{C}_6$ and its generalization Theorem $\mathcal{C}^∗_6$, which cover the “$p$-intermediate” case. $\ldots$ In the preceding book in this series, we had promised complete proofs of Theorems $\mathcal{C}_6$ and $\mathcal{C}^∗_6$ in this book, but because of space considerations, we postpone the completion of those theorems to the next volume.
EDIT September 2021
In response to a question from Hugo de Garis, Ron Solomon sent the following email in January 2021:
Vol. 9 is already submitted, accepted and scheduled for publication. It should be published early this year.
As for the rest, my best guess now is that there will in fact be 4 further volumes, not 3. A roughly 800 pages manuscript on the Uniqueness Theorem has been completed by Gernot Stroth. With some additional material, it will fill 2 further volumes. This could probably be readied for publication by a year from now. However, our team (Inna Capdeboscq, Richard Lyons, Chris Parker and myself) are currently focussing on the remaining work to be done for the other two volumes. It is difficult to estimate how long this will take. With luck we might have a first draft completed this calendar year, but it might take longer.
It is safe to say that the remaining volumes will not all be published before 2023. I hope it is also safe to say that they will all be published no later than 2025.
(Emphasis added)
EDIT 29 Dec 2021
Richard Lyons maintains an erratum for the whole published second generation CFSG on this page: https://sites.math.rutgers.edu/~lyons/cfsg/
EDIT 05 Apr 2022
In response to a further question from de Garis (see the page linked above), Solomon wrote (in March 2022):
We have been working on the theorems for both Volumes 10 and 11. Just in the past few weeks, we have decided to concentrate on the completion of Volume 10. This is proceeding very well and we should be able to submit Volume 10 for publication this year, I believe. I fear that I may have been a bit overoptimistic in predicting the completion of all the volumes by the end of 2024.
Here's some information from Barry Cipra's June 1988 article "Fermat's Theorem remains unproved" in Science magazine.
Parshin showed that the arithmetical version of a certain inequality involving geometric invariants of surfaces—an inequality that Miyaoka proved for the geometric case in 1974—would lead by a series of steps to a bound on the size of possible exponents for which Fermat's Last Theorem could be false. $\ldots$
Miyaoka's work is directed at proving the arithmetical inequality. Miyaoka, who is an expert in algebraic geometry but a relative newcomer to the arithmetical theory, proceeded by analogy with the geometric case. But according to Enrico Bombieri, a professor of mathematics at the Institute for Advanced Studies [sic] in Princeton, the translation is not straightforward. "Things go over, but with some qualifications," Bombieri says. "The naïve extension doesn't go through."
The problem, according to Barry Mazur of Harvard University, is the lack of a good arithmetical analog of a crucial geometric object known as the tangent bundle. Mazur, who helped Miyaoka analyze the proof, explains that Miyaoka had "a very interesting idea" to replace the tangent bundle with a "generic" bundle, with the assumption that the generic bundle can be chosen so as to have suitably nice properties. This seems not to be the case.
The effort is not wasted, however, Mazur says that Miyaoka has carried the idea of substituting generic bundles for the tangent bundle back to the original geometric case. "Given any choice of a bundle, you'll get some inequalities," Mazur says. "It's a perfectly reasonable and interesting geometric question to ask what's the structure of this whole complex set of inequalities." Answering such questions will very possibly lead to a deeper understanding of Miyaoka's original geometric proof.
More information about how the inequality in question (known as the Bogomolov–Miyaoka–Yau inequality) relates to Fermat's Last Theorem can be found in the appendix (by Paul Vojta) to Serge Lang's book Introduction to Arakelov Theory.
Best Answer
Here is an answer from my point of view, immersed as I am -- Geoff is right -- in the second generation project. First, a few general comments. Our overriding purpose has been to expound a coherent proof of CFSG that is supported completely by what we call ``Background Results,'' an explicit and restricted list of published books and papers, plus the assertion that every one of the $26$ sporadic groups is determined up to isomorphism, as a finite simple group, by its so-called centralizer-of-involution pattern. This list has changed over the years. In our first volume it is explicitly listed as we conceived it at the time (1990's). Further additions, mostly of post-first-generation publications, are noted as they have been adopted in subsequent volumes. (Some of these additions are characterizations of some sporadic groups--for example, the Monster and Baby Monster--by weaker data than centralizer-of-involution pattern, so that they supplant the earlier Background Results characterizing those groups.) The biggest additions, by far, are Aschbacher and Smith's monumental books on the quasi-thin problem, since we were hardly going to do it as well ourselves, let alone better. Whatever errors may be in the second-generation proof, therefore, are either in the Background Results or in our series.
Naturally, we have taken ideas and arguments from many papers and books outside the Background Results in formulating our proof. Occasionally in the course of understanding these results, or adapting them for our purposes, we have uncovered gaps. None of these is at all comparable in scope (by orders of magnitude) to the well-known quasi-thin gap that Aschbacher and Smith bridged; in that sense, they could be called ``minor.'' To deal with these gaps, when they threatened our proof, we have either found alternative arguments ourselves, or asked the authors for help. In every case, so far, the gap has been closed in one of these two ways. However, and unfortunately for the purposes of answering your question, we have not kept a log of these incidents. Nor have we by any means intended to examine every paper needed in the first-generation proof this way. We are guided just by what we need in the second generation.
Here is an example of a minor gap that came to our attention in the preparation of volume $9$. We needed a certain characterization of the $7$- and $8$-dimensional orthogonal groups over the field of $3$ elements. We were guided by an important paper by Aschbacher that had appeared relatively late and without much fanfare in the first generation. There was an apparent gap -- very technical -- in the paper, and Professor Aschbacher promptly supplied us with a correction.
Another example that I know well, from before the CFSG, came in $1972$ in my paper pointing to the possible existence of the sporadic group $Ly$. I asserted that if such a group existed, then every nonidentity element of order a power of $5$ actually would have order $5$. Koichiro Harada wrote me shortly thereafter that on the contrary, there would be elements of order $25$. He was right; I had miscalculated. Luckily, the miscalculation did not affect the rest of the paper.