The monomial Burnside functor

Given a finite group G, we can realize the permutation modules by the linearization map defined from the Burnside ring B(G) to the character ring of G, denoted AK(G). But not all KG-modules are permutation modules. To realize all the KGmodules we need to replace B(G) by the monomial Burnside ring BC(G). We can get information about monomial Burnside ring of G by considering subgroups or quotient groups of G. For this the setting of biset functors is suitable. We can consider the monomial Burnside ring as a biset functor and study the elemental maps: transfer, retriction, inflation, deflation and isogation. Among these maps, deflation is somewhat difficult and requires more consideration. In particular, we examine deflation for p-groups and study the simple composition factors of the monomial Burnside functor for 2-groups with the fibre group {±1}.