Difference Sets Disjoint from a Subgroup

We study finite groups G having a non-trivial, proper subgroup H and D ⊂ G \ H , D ∩ D - 1 = ∅ , such that the multiset { x y - 1 : x , y ∈ D } has every non-identity element occur the same number of times (such a D is called a difference set ). We show that | G | = | H | 2 , and that | D ∩ H g | = | H | / 2 for all g ∉ 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.