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][1]).

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


  [1]: https://mathoverflow.net/questions/103368/automorphisms-of-quotient-groups