Direct product of automorphism groups of digraphs

We study the direct product of automorphism groups of digraphs, where automorphism groups are considered as permutation groups acting on the sets of vertices. By a direct product of permutation groups $(A,V)times (B,W)$ we mean the group $(Atimes B,Vtimes W)$ acting on the cartesian product of the respective sets of vertices. We show that, except for the infinite family of permutation groups $S_n times S_n$, $n geq 2$, and four other permutation groups, namely $D_4 times S_2$, $D_4 times D_4$, $S_4 times S_2 times S_2$, and $C_3 times C_3$, the direct product of automorphism groups of two digraphs is itself the automorphism group of a digraph. In the course of the proof, for each set of conditions on the groups $A$ and $B$ that we consider, we indicate or build a specific digraph product that, when applied to the digraphs representing $A$ and $B$, yields a digraph whose automorphism group is the direct product of $A$ and $B$.