This concerns one of those "well known" facts, referred to in a recent preprint I've been looking at. In principle it's elementary, but I can't pin down an explicit textbook reference for it. Start with two finite groups $A,B$ and their product $G:=A \times B$, working over a splitting field $K$ for the groups involved with prime characteristic dividing $|G|$. Let $S_1, \dots, S_m$ and $T_1, \dots, T_n$ be respective sets of representatives of isomorphism classes of simple modules for the group algebras $KA, KB$. In turn let the projective covers (=injective hulls) be respectively $P_i, Q_j$. These are the PIMs or indecomposable projective modules for the two group algebras.

It's a standard observation (found in some books) that there is an obvious isomorphism between $KG$ and the tensor product algebra $KA \otimes_K KB$, while each group algebra splits into the direct sum (as a left module over itself) of the various PIMs taken with multiplicity equal to the dimension of the corresponding simple module. It's also a standard fact (found in some books) that each $S_i \otimes T_j$ is a simple module for $KG$. From these ingredients one can conclude that $P_i \otimes Q_j$ is the corresponding PIM, thereby exhausting all isomorphism classes for $KG$.

Is all of this written down in a self-contained way somewhere?