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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.

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    $\begingroup$ I think it very unlikely that there are easily stated necessary and sufficient conditions for this property. $\endgroup$
    – Derek Holt
    Commented Oct 4, 2023 at 7:40
  • $\begingroup$ @DerekHolt Oh, I understand, is there any literature on this? I could not find it, that is why I am asking. $\endgroup$
    – Ratanjit
    Commented Oct 4, 2023 at 8:25
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    $\begingroup$ You would not expect there to be literature if there was nothing much to say about it. The question is too general. If you were interested in more specific instances of the problem then you might be able to say more. For example if $G/N$ is perfect and $G$ is its covering group then the answer is yes. $\endgroup$
    – Derek Holt
    Commented Oct 4, 2023 at 9:14
  • $\begingroup$ @DerekHolt I got it, thank you. $\endgroup$
    – Ratanjit
    Commented Oct 4, 2023 at 9:50

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

Also here on math.SE:

A concrete counterexample can be found here:

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

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