Wreath product in automorphism groups of graphs

The automorphism group of the composition of graphs G circle H $G\circ H$ contains the wreath product A u t ( H ) wreath product A u t ( G ) $Aut(H)\,\wr \,Aut(G)$ of the automorphism groups of the corresponding graphs. The classical problem considered by Sabidussi and Hemminger was under what conditions G circle H $G\circ H$ has no other automorphisms. In this paper we consider questions related to the converse: if the automorphism group of a graph is a wreath product A wreath product B $A\,\wr \,B$, are the smaller groups necessarily automorphism groups of graphs? And if so, are the corresponding smaller graphs involved in the construction? We consider these questions for the wreath product in its natural imprimitive action (which refers to the results by Sabidussi and Hemminger), and in generalization to colored graphs, which seems to be a more appropriate setting. For this case we have a fairly complete answer. Yet, we also consider the same problems for the wreath product in its product action. This turns out to be more complicated and we have only partial results. Our considerations in this part lead to interesting open questions involving hypergraphs and to an analogue of the Sabidussi-Hemminger problem for a related graph construction.