Does there exist a group $G$ such that ${\rm Aut}(G) = S_6$?

Question

I thought this would be an interesting question, since every other symmetric group on $n$ elements is possible as an automorphism group, since ${\rm Aut}(S_n) \cong S_n$ for $n \neq 2, 6$. Obviously, $S_2 \cong C_2$ is a possibility... So is $S_6$ the automorphism group of any finite group? Sorry if this is a silly question.