On permutation characters of finite group

Let $G$ be a finite group and $M$ be a maximal subgroup of $G$. We call every irreducible constituent $\chi$ of $(1_M{)}^G$ a $\mathcal{P}$-character of $G$ with respect to $M$. In this paper, we prove that all $\mathcal{P}$-characters of $G$ are monomial if and only if $G$ is solvable, which solves a question posed by Qian and Yang.