Let $G$ a finite abelian group and $M$ a finitely generated G-module.
We consider the cohomology module $H^*(G;M)$, which is a $H^*(G;\mathbb{Z})$-module with the cup-product.
Is always $H^*(G;M)$ finitely generated? Or, at least, is there any condition which on $M$ or $G$ which ensures it?