It is known that a Cayley digraph Cay(A, S) of an abelian group A is isomorphic to a nontrivial wreath product if and only if there is a proper nontrivial subgroup B≤A such that S\B is a union of cosets of B in A. We generalize this result to Cayley graphs Cay(G, S) of nonabelian groups G by showing that such a digraph is isomorphic to a nontrivial wreath product if and only if there is a proper nontrivial subgroup H≤G such that S\H is a union of double cosets of H in G. This result is proven in the more general situation of a coset digraph. We will also discuss implications of this result to coset digraphs. This is joint work with Rachel Barber of Mississippi State University.