Generalized Burnside rings and group cohomology

We define the cohomological Burnside ring Bn(G,M) of a finite group G with coefficients in a ZG-module M as the Grothendieck ring of the isomorphism classes of pairs [X,u] where X is a G-set and u is a cohomology class in a cohomology group . The cohomology groups are defined in such a way that when X is the disjoint union of transitive G-sets G/Hi. If A is an abelian group with trivial action, then B1(G,A) is the same as the monomial Burnside ring over A, and when M is taken as a G-monoid, then B0(G,M) is equal to the crossed Burnside ring Bc(G,M). We discuss the generalizations of the ghost ring and the mark homomorphism and prove the fundamental theorem for cohomological Burnside rings. We also give an interpretation of B2(G,M) in terms of twisted group rings when M=k× is the unit group of a commutative ring.