On some saturated triples and hyperfocal subgroups

Let b denote a primitive idempotent of the subalgebra (kH)G in kH of G-stable elements, where H is a normal subgroup of a finite group G which acts by conjugation on H, and k is an algebraically closed field of characteristic p. Using a generalization of block induction to the case of primitive idempotents in algebras of the form (kH)G we show that, when the hyperfocal subgroup is abelian, the fusion system associated to b, respectively to the induced stable primitive idempotent of the generalized Brauer pair corresponding to the hyperfocal subgroup, are the same.