Let $Aut(G)$ be the group of automorphisms of $G$. Does there exist a bigger group $X$ such that $G < X$ and, for every $\tau \in Aut(G)$, there exists $\phi \in Inn(X)$ such that $\tau=\phi|_G$, where $Inn(X)$ is the group of inner automorphism of $X$ and $\phi|_G$ is the restriction of $\phi$ to $G$?
This question happened to me while I was reading the fact that every characteristic subgroup of a normal subgroup $N$ is normal in the whole group $G$. The proof of this comes from the fact that the restriction of every inner automorphism of $G$ to $N$ is an automorphism in $N$ so the characteristic subgroup is invariant under all inner automorphism of $G$.