A filtration of the modular representation functor

Let F and K be algebraically closed fields of characteristics p > 0 and 0, respectively. For any finite group G we denote by K R_F (G) = K ⊗_Z G₀ (F G) the modular representation algebra of G over K where G₀ (F G) is the Grothendieck group of finitely generated F G-modules with respect to exact sequences. The usual operations induction, inflation, restriction, and transport of structure with a group isomorphism between the finitely generated modules of group algebras over F induce maps between modular representation algebras making K R_F an inflation functor. We show that the composition factors of K R_F are precisely the simple inflation functors S_{C, V}ⁱ where C ranges over all nonisomorphic cyclic p^′-groups and V ranges over all nonisomorphic simple K Out (C)-modules. Moreover each composition factor has multiplicity 1. We also give a filtration of K R_F.