TITLE: Finding the outer automorphism group just by using the automorphism group
QUESTION [5 upvotes]: Let $G$ be a group and let $H$ be its automorphism group $H = Aut(G)$. However, suppose that we did not have knowledge of $G$. Is there a way to find the outer automorphism group of $G$, $Out(G)$ just by using $H$? 
Thanks

REPLY [4 votes]: No. For example: If $n>6$, then $S_n\simeq Aut(S_n)\simeq Aut(A_n)$, so the possibilities $Out(G)=Out(S_n)=1$ and $Out(G)=Out(A_n)=C_2$ both remain.