Is there any nontrivial group that is isomorphic to its outer automorphism group

Question

I know that there are lots of nontrivial groups that are isomorphic to their automorphism groups like $S_3$. Is there any nontrivial group that is isomorphic to its outer automorphism group? Is there any nontrivial finite group isomorphic to both its inner automorphism group and its outer automorphism group? If so, what is the classification of such groups?

Answer 1

There is a famous result, that the outer automorphism group of a finite simple group is solvable, see the Schreier conjecture. In particular, it cannot be isomorphic to the group itself.

It is interesting to note that the corresponding question for Lie algebras, posed by Hans Zassenhaus, is different, i.e., there are indeed simple Lie algebras isomorphic to their outer derivation algebra, namely for example the simple Lie algebra $\mathfrak{psl}_3(\Bbb F)$ of dimension $7$ over a field $\Bbb F$ of characteristic three.

For solvable groups $G$, it can easily happen that $G\cong Out(G)$, e.g., for $G=C_2\times D_4$, as noted by Derek. There are some more posts related to this topic, or similar topics, for example

Schreier's Conjecture

Is every finite group the outer automorphism group of a finite group?

Group isomorphic to its automorphism group