@FernandoMuro I thought it was an open problem (generating hypothesis) if we assume only that $\pi_{\ast}(f)=0$...

Thanks for all the details! Do you think that if we add the condition that $M$ is finitely generated as $\mathbb{Z}G$-module, then the conclusion will be that $M=0$ ? Since the kernel of $\mathbb{Z}G\rightarrow \mathbb{Z}$ is not finitely generated (hope I am not wrong)