Multiple Burnside rings and Brauer induction formulae

Let G be a finite group, and suppose that G is an operator group of a finite group A. Define , where S(G) is the set of subgroups of G and Z1(H,A) is the set of crossed homomorphisms from H to A. We view G as an operator group of the opposite group A○ of A, and make S(G,A) into a left A○⋊G-set. The ring Ω(G,A) is defined to be a commutative ring consisting of all formal Z-linear combinations of A○⋊G-orbits in S(G,A). Idempotent formulae for Q⊗ZΩ(G,A) not only imply a generalization of Dress' induction theorem but bring, in the case where Z1(G,A) is the set of linear C-characters of G, Boltje's explicit formula for Brauer's induction theorem and its hyperelementary version.