= Solution
Let $D,E$ be <module vertices> of $M$, and choose a <module> $T$ such that $M\mid\operatorname{Ind}_E^G T$. Relative $D$-projectivity then gives
$$
M\mid\operatorname{Ind}_D^G\operatorname{Res}_D^G\operatorname{Ind}_E^G T.
$$
Apply <Mackey decomposition> to the inner restriction and use induction transitivity. The resulting <module> is a <direct sum> of <modules> induced to $G$ from <subgroups> $D\cap{}^gE$, 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 $D=D\cap{}^gE$, so $D\le{}^gE$.
Interchanging $D,E$ gives $E\le{}^hD$ for some $h$. The two inclusions force equal orders, and hence
$$
\boxed{D={}^gE.}
$$
Thus all <vertices of an indecomposable module> are conjugate in $G$.
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