Difference Sets Disjoint from a Subgroup

We study finite groups G having a non-trivial, proper subgroup H and D subset of G\H, D boolean AND D-1 = emty set, such that the multiset {xy(-1) : x, y is an element of D} has every non-identity element occur the same number of times (such a D is called a difference set). We show that vertical bar G vertical bar = vertical bar H vertical bar(2), and that vertical bar D boolean AND Hg vertical bar = vertical bar H vertical bar/2 for all g is not an element of H. We show that H is contained in every normal subgroup of index 2, and other properties. We give a 2-parameter family of examples of such groups. We show that such groups have Schur rings with four principal sets, and that, further, these difference sets determine DRADs.