Exercise V.2.11, on p 320 of Kunen's Set Theory (the newer one) states
Starting with $\sf V=L$ in the ground model, follow the suggestion of
Exercise II.8.12 and obtain a generic extension satisfying the
statements $(2^{\aleph_0}=\aleph_2)^{\sf HOD}$ and $\sf L=
HOD^{HOD}\subsetneq HOD\subsetneq V.$Hint You can use an automorphism argument, as in Section IV.8 to prove that a set is not in $\sf HOD.$ Also, the "zero-one" law
(Theorem IV.4.15) shows that if you start with $\sf V=L$ and force
with $\operatorname{Fn}(\omega_2,2),$ the generic extension will
satisfy $2^{\aleph_0}=\aleph_2$ and $\sf HOD=L.$
The exercise (II.8.12) that the question says to follow reads:
Working in $\sf ZFC,$ let $C_\alpha=\{n\in \omega:
> 2^{\aleph_{\alpha+n}}=\aleph_{\alpha+n+1}\}.$ Assume that
$|\{C_\alpha: \alpha\in \sf Ord\}|\ge \aleph_2.$ Prove that $\sf CH$
is false in $\sf HOD.$
There are a couple references in the hint that I won't reproduce – I think what he's referring to with "automorphism argument" should be apparent and the "zero-one" law just says that for any sentence $\varphi$, $1\Vdash \varphi $ or $1\Vdash \lnot \varphi$ when $P$ is weakly homogeneous.
The hint and the referenced exercise seem to suggest that we first add $\omega_2$ Cohen reals to $\sf L$ to produce a model $M$ with $M\models \sf HOD = L$ and $2^{\aleph_0}=\aleph_2$ (that all checks out) and then we force again with an Easton product to make a model $N$ satisfying something like $2^{\aleph_{\alpha+n}}=\aleph_{\alpha+n +1}$ if $n\in C_\alpha$ and $2^{\aleph_{\alpha+n}}=\aleph_{\alpha+n +2}$ if $n\notin C_\alpha$, where $\{C_\alpha :\alpha <\omega_2\}$ is an enumeration of $P(\omega)$ in $M.$ (This would need to be adjusted to avoid singular cardinals, but don't want to get too hung up on details.)
I would hope then the idea is to show that $\mathsf{HOD}^N = M,$ since that resolves the problem rather nicely, but I don't see how. The coding into the GCH pattern makes it so all the reals in $M$ are in $\mathsf{HOD}^N,$ but I don't see how we can extend that to all of $M.$
In the comments of this question I asked a while ago, Noah helpfully pointed out the Fuchs/Hamkins/Reitz set theoretic geology paper that shows for any model $V$ there's a forcing extension such that $V=\mathsf{HOD}^{V[G]}$ (though they show a lot more and I gather the narrow result goes back much further than that, and that it can probably be simplified). This solves this problem, but it uses class forcing (which Kunen hasn't covered yet) and doesn't have much resemblance to Kunen's exercise other than that they both use the GCH pattern.
So does anybody have any insight on what Kunen had in mind here? My only guess is that since $M$ is given by such a mild forcing from $L$ that we somehow can avoid worrying about stuff further up.
Edit
I may have asked this question too early. After a little more thought it seems obvious that my last remark is onto something since the sets in $M=L[G]$ are all ordinal-definable from $G$, so if we have successfully coded $G$ into the continuum function in $N$, we should be able to deduce that $M\subseteq \mathsf{HOD}^N.$
Would still appreciate some advice on clinching this, as well as clinching $\mathsf{HOD}^N\subseteq M,$ if indeed these things are true.
Best Answer
Yes. Your idea is fine.
If we start with $L$, add $\aleph_2$ Cohen reals, with $L[G_0]$ being the generic extension, then by homogeneity ${\rm HOD}^{L[G_0]}=L$, and we can now encode $G_0$ into $\rm HOD$ by many kind of ways. For example, by forcing with a product of $\operatorname{Add}(\omega_\xi,1)^L$ for an appropriate choice of $\xi$, or by collapsing successor cardinals, or by increasing the continuum at those points, etc. The point is that we can identify for each $\alpha$ an interval of length $\omega$ which looks like the $\alpha$th Cohen real. One key point of importance is that we need the forcing to be homogeneous, which, all of those examples provide.
Now, if $G_1$ was the $L[G_0]$-generic filter for this encoding forcing, we get that ${\rm HOD}^{L[G_0][G_1]}$ is contained in ${\rm HOD}(\Bbb P)^{L[G_0]}$, where $\Bbb P$ was the forcing used for the encoding. Since $G_0$ and $\Bbb P$ are constructible from each other by design, this is exactly $L[G_0]$. And so we get that in $L[G_0][G_1]$ the following holds: $L={\rm HOD^{HOD}\subsetneq HOD}\subsetneq V$.