 Solution

Let $D,E$ be module vertices of $M$, and choose a module $T$ such that $M\mid {\mathrm{Ind}}_E^{G}T$. Relative $D$-projectivity then givesApply Mackey decomposition to the inner restriction and use induction transitivity. The resulting module is a direct sum of modules induced to $G$ from subgroups $D\cap {}^{g}E$, for representatives $g\in D\backslash G/E$. By Krull–Schmidt theorem, the indecomposable module $M$ is a summand of one of these induced representations. It is therefore relatively projective for one intersection contained in $D$. Minimality of $D$ forces $D=D\cap {}^{g}E$, so $D\le {}^{g}E$.

Interchanging $D,E$ gives $E\le {}^{h}D$ for some $h$. The two inclusions force equal orders, and henceThus all vertices of an indecomposable module are conjugate in $G$.