I am concerning here a natural question:

$G$ 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).

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