Note: Let $G$ be a finite group and $H \leq G$. Then it is clear that if $H \unlhd G$, then $\chi(h) \neq 0$ for irreducible constituents $\chi$ of the permutation character $(1_H)^G$ and $h\in H$. The converse does not hold in general.
Let $G$ be a finite group and $P\in \mathrm{Syl}_p(G)$. Then it was shown by Navarro in (1) that $P \unlhd G$ if and only if for all irreducible constituents $\chi$ of the permutation character $(1_P)^G$ and all $x\in P$, that $\chi(x) \neq 0$.
He then followed up this idea in (2), to show that if $G$ is a finite solvable group with a pronormal subgroup $H$ of $G$. Then $H \unlhd G$ if and only if $\chi(h) \neq 0$ for irreducible constituents $\chi$ of $(1_H)^G$ and $h\in H$. It was also demonstrated that the solvability condition could not be dropped.
Another result in (2) showed that if $G$ is a finite nilpotent group and $H \leq G$, then $H \unlhd G$ if and only if $\chi(h) \neq 0$ for irreducible constituents $\chi$ of $(1_H)^G$ and $h\in H$.
I was wondering what other families of subgroups one could explore such that the converse of the note above holds. Any insight will be welcomed.
