I am concerning here a natural question:

> **Problem:** $G$ is a finite group, and $N$ is ca characteristic subgroup. Given a single $\varphi\in\mathrm{Aut}(G/N)$, when it can 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 $N$, this Problem has been studied?


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