Indecomposable modules for direct products of finite groups

An essentially known result is made explicit and its converse is proved, thereby showing that if $K$ is a field of prime characteristic $p$, $P$ a finite $p$-group and $H$ a finite $p'$-group, then every finitely generated indecomposable $K(P\times H)$-module is a tensor product of an indecomposable $KP$-module with an indecomposable $KH$-module if and only if either $P$ is cyclic or $K$ is a splitting field for $H$.