  We define a corestriction map for equivariant Brauer groups in the sense of
Fr\"ohlich and Wall, which contain as a special case the Brauer-Clifford groups
introduced by Turull. We show that this corestriction map has similar
properties as the corestriction map in group cohomology (especially Galois
cohomology). In particular, composing corestriction and restriction associated
to a subgroup $H\leq G$ amounts to powering with the index $\lvert G:H \rvert$.
