Let $G$ be a finite M-group (ie, where all irreducible complex characters are induced from linear characters of subgroups). What subgroups of $G$ are necessarily themselves M-groups? For instance, clearly any nilpotent subgroup will be an M-group (so the Fitting subgroup works), and it has been shown that all normal Hall-subgroups are themselves M-groups. On the other hand, not all normal subgroups need to be M-groups, as has been shown by Dade, though it seems like most of them will be.

For instance, is it known whether the derived subgroup of an M-group is itself an M-group?

My reason for asking is that I am looking at a class of groups that behaves to some extent like M-groups, and it looks like all M-groups might indeed be in this class, but to show this it would be very helpful to know some subgroups (not necessarily normal) that are guaranteed to be M-groups themselves.

Edit: Given an irreducible character $\chi$, we can look at the set of subgroups of $G$ from which $\chi$ is induced from a linear character. Does anyone know an example of an M-group $G$ and an irreducible character $\chi$ of $G$ such that this set of subgroups does not contain any M-groups?