Let $G$ be an M-group. Let $\chi$ be an irreducible character of $G$ and set $$A_{\chi} = \{ H\leq G\ |\ \exists \psi\in\textrm{Lin}(H): \psi^G = \chi\}$$ Since $G$ is an M-group, this set is non-empty for each $\chi\in\textrm{Irr}(G)$ (this is one way to define M-groups). Is it known whether $A_{\chi}$ always contains an M-group, or are there conditions known on $\chi$ that guarantee this (apart from $\chi$ being linear)?

Note: A while ago I asked a question (What subgroups of M-groups are guaranteed to be M-groups themselves?) about what subgroups of M-groups that are themselves guaranteed to be M-groups. That question was probably too general to have a nice answer for the time being, and my edit to ask a more specific question may have been lost in the question itself.