When the natural homomorphism from $Aut(G)$ to $Aut(G/N)$ is onto?

Question

For the last few days, I have been working on a bunch of group theory problems and I am stuck with one question that came into my mind. My level is up to an introductory course in group theory and maybe the answer to my question is known, but I do not know where to find its solution. My question is the following:

Let $N$ be a characteristic subgroup of a group $G$. We know that if $\sigma: G \rightarrow G$ is an automorphism then there is an automorphism $\sigma^{\ast}: G/N \rightarrow G/N$ given by $gN \mapsto \sigma(g)N$. Thus, we have an homomorphism $\rho: Aut(G) \rightarrow Aut(G/N)$. My question is when the homomorphism $\rho$ is surjective. Is there a necessary and sufficient condition so that $\rho$ is surjective? Please help me, I am not able to solve this question.

Answer 1

Derek is right, nothing really can be said here in general. The same question has been asked twice on MO a few years ago:

• https://mathoverflow.net/questions/235646/lifting-automorphisms-of-quotient-groups
• https://mathoverflow.net/questions/103368/automorphisms-of-quotient-groups

Also here on math.SE:

• Automorphisms inducing automorphisms of quotient groups

A concrete counterexample can be found here:

• Lifting an isomorphism of quotients to an automorphism of free group

The paper Lifting Automorphisms of Quotients by Central Subgroups by Ben Kane and Andrew Shallue deals with the case that $N$ is central.

• https://arxiv.org/pdf/1304.4632.pdf