On Clifford theory with Galois action

Let Gˆ be a finite group, N   a normal subgroup of Gˆ and ϑ∈IrrN. Let FF be a subfield of the complex numbers and assume that the Galois orbit of ϑ   over FF is invariant in Gˆ. We show that there is another triple (Gˆ1,N1,ϑ1) of the same form, such that the character theories of Gˆ over ϑ   and of Gˆ1 over ϑ1ϑ1 are essentially “the same” over the field  FF and such that the following holds: Gˆ1 has a cyclic normal subgroup C   contained in N1N1, such that ϑ1=λN1ϑ1=λN1 for some linear character λ of C  , and such that N1/CN1/C is isomorphic to the (abelian) Galois group of the field extension F(λ)/F(ϑ1)F(λ)/F(ϑ1). More precisely, having “the same” character theory means that both triples yield the same element of the Brauer–Clifford group BrCliff(G,F(ϑ))BrCliff(G,F(ϑ)) defined by A. Turull.