Let $G$ be a group, with $N$ characteristic in $G$. As $N$ is characteristic, every automorphism of $G$ induces an automorphism of $G/N$. Thus, $\operatorname{Aut}(G)\rightarrow \operatorname{Aut}(G/N)$. I was therefore wondering,
Under what conditions is the induced homomorphism $\operatorname{Aut}(G)\rightarrow \operatorname{Aut}(G/N)$
a monomorphism?
an epimorphism?
an isomorphism?
I believe it should work for (semi-?)direct products $N\times H$ where $\operatorname{Aut}(N)$ is trivial and $N\not\cong H$ (for example, $C_2\times C_3$, $N=C_2$). But I can't prove even that!