Completing an extension of Tunnell's theorem

We look at a special case of a familiar problem: Given a locally compact group G, a subgroup H and a complex representation π+ of G how does π+ decompose on restriction to H. Here G is GL+(2,F), where F is a nonarchimedian local field of characteristic not two, K a separable quadratic extension of F, GL+(2,F) the subgroup of index 2 in GL(2,F) consisting of those matrices whose determinant is in NK/F(K∗), π+ is an irreducible, admissible supercuspidal representation of GL+(2,F) and H=K∗ under an embedding of K∗ into GL(2,F).