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I am concerning here a natural question:

Problem: Let $G$ be a finite group, and let $N$ be a characteristic subgroup of $G$. When can an automorphism $\varphi\in\mathrm{Aut}(G/N)$ be lifted to an automorphism of $G$?

The problem in general seems difficult, and it has raised four years before in MathOverflow (see this).

Question: for which classes (or cases) of groups $G$ or $N$ has this Problem been studied?

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    $\begingroup$ Consider the case that $N = \Phi(G)$ in which case $G/N$ has large automorphism group. There are many cases though when ${\rm Aut}(G)$ is a $p$-group. $\endgroup$
    – Geoff Robinson
    Commented Apr 8, 2016 at 6:21
  • $\begingroup$ The question as stated doesn't seem to admit a unique correct answer. A better, more focused, question, would ask whether the question has been studied in a particular instance. $\endgroup$
    – HJRW
    Commented Apr 8, 2016 at 8:41
  • $\begingroup$ The question also seems to have changed. My comment above concerned the case $N$ a p-group. $\endgroup$
    – Geoff Robinson
    Commented Apr 8, 2016 at 12:52

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