Linking conjugacy classes and minimal invariant characters of normal subgroups

Let $G$ be a finite group and $N$ a normal subgroup of $G$. We report on recent results concerning minimal $G$-invariant characters of $N$ (which are the sums of the characters on each orbit of the action of $G$ by conjugation on $\text{Irr}(N)$) and their influence on the structure of $N$, as well as their relationship to the $G$-conjugacy classes of $N$.