Let $\alpha\colon G\to H$ be a group homomorphism and $R$ be a commutative ring. Regard $R$ as $RN$-module with trivial $N$-action.
Further, we denote $N=\operatorname{Ker}\alpha$.
Is it true that $\mathbb{Z}G\otimes_{\mathbb{Z}}R\otimes_{RN}R=RH$ even $\alpha$ is not onto?
