Let $G = X \ltimes H$ be a finite group, semidirect product with normal divisor $H$. Let $K$ be a field over which irreducible modules are totally irreducible for $G$. Let the Norm be $N = \sum_{g \in H} g$ and let $J = ( 1 - \frac{N}{| H |})$ be the idempotent that generates the augmentation submodule $A := J . K[ G ] \subset K[ G ]$. Let $M \subset A$ be a $(K[ G ]-K[ H ])$ bimodule.
Is it in general true that every submodule $M^\prime \subset M$ is a bilateral ideal? This holds for metabelian $G$, but for the general case the proof may be technically more involved, and I wonder if it can follow from more general results on induction, for instance?