I have a proof of the following result but I was wondering if anyone had a reference for it. I have asked on math.stackexchange here but didn't receive any replies.

Let $G$ a finite group given by the (inner) central product of two subgroups $H$ and $K$ over $M$ (I am using the definition of central product given in Gorenstein's "Finite groups"; in particular $G=HK$ and $H \cap K=M \subset Z(G)$). If R is a commutative ring, then $R[G]=R[H] \otimes_{R[M]} R[K]$.