The Schur–Clifford subgroup of the Brauer–Clifford group

We define a Schur–Clifford subgroup of Turull's Brauer–Clifford group, similar to the Schur subgroup of the Brauer group. The Schur–Clifford subgroup contains exactly the equivalence classes coming from the intended application to Clifford theory of finite groups. We show that the Schur–Clifford subgroup is indeed a subgroup of the Brauer–Clifford group, as are certain naturally defined subsets. We also show that this Schur–Clifford subgroup behaves well with respect to restriction and corestriction maps between Brauer–Clifford groups.