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?

${\rm S}_6$ is not the automorphism group of a finite group.
See H.K. Iyer, On solving the equation Aut(X) = G, Rocky Mountain J. Math. 9 (1979), no. 4, 653--670, available online
here.

This paper proves that for any finite group $G$, there are finitely many 
finite groups $X$ with ${\rm Aut}(X) = G$, and it explicitly solves the
equation for some specific values of $G$.
In particular, Theorem 4.4 gives the complete solution for $G$
a symmetric group, and when $n = 6$ there are no such $X$.
