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?