Which Haar graphs are Cayley graphs?

For a finite group G and subset S of G, the Haar graph H(G,S) is a bipartite regular graph, defined as a regular G-cover of a dipole with |S| parallel arcs labelled by elements of S. If G is an abelian group, then H(G,S) is well-known to be a Cayley graph; however, there are examples of non-abelian groups G and subsets S when this is not the case. In this paper we address the problem of classifying finite non-abelian groups G with the property that every Haar graph H(G,S) is a Cayley graph. An equivalent condition for H(G,S) to be a Cayley graph of a group containing G is derived in terms of G,S and AutG. It is also shown that the dihedral groups, which are solutions to the above problem, are Z2, D3, D4 and D5.