Let $G$ be a finite group, $H$ a maximal subgroup. If $[G:H] = 2$, it is very well known how to determine the conjugacy classes of elements of $H$: they either stay the same or split depending on whether the representatives are centralized by some element outside of $H$.
Can this argument be generalized to a maximal subgroup which is normal, but not necessarily of index 2? Can it be generalized to any maximal subgroup? The last one might be tricky because the $G$-conjugacy class is not necessarily contained in $H$.