TITLE: Is ${\rm S}_6$ the automorphism group of a group?
QUESTION [72 upvotes]: The automorphism group of the symmetric group $S_n$ is $S_n$ when $n$ is not $2$ or $6$, in which cases it is respectively $1$ and the semidirect product of $S_6$ with the (cyclic) group of order $2$. (For this famous outer automorphism, see for instance wikipedia or Baez's thoughts on the number $6$.)
On the other hand, $S_2$ is the automorphism group of $Z_3$, $Z_4$ and $Z_6$ (and only those groups among finite groups). Hence my question: is $S_6$ the automorphism group of a group? of a finite group?

REPLY [7 votes]: There is a whole array of results, going back to G. Birkhoff at 1930s saying that every group is an automorphism group of some universal algebra (or some universal algebra inside some class).

(This really should be merely a comment to the previous answer, but I am still not reputable enough to leave comments).