Composition factors of functors

Let X be a family of finite groups satisfying certain conditions and K be a field. We study composition factors, radicals, and socles of biset and related functors defined on X over K. For such a functor M and for a group H in X, we construct bijections between some classes of maximal (respectively, simple) subfunctors of M and some classes of maximal (respectively, simple) KOut(H)-submodules of M(H). We use these bijections to relate the multiplicity of a simple functor SH,V in M to the multiplicity of V in a certain KOut(H)-module related to M(H). We then use these general results to study the structure of one of the important biset and related functors, namely the Burnside functor BK which assigns to each group G in X its Burnside algebra BK(G)=K⊗ZB(G) where B(G) is the Burnside ring of G. We find the radical and the socle of BK in most cases of X and K. For example, if K is of characteristic p>0 and X is a family of finite abelian p-groups, we find the radical and the socle series of BK considered as a biset functor on X over K. We finally study restrictions of functors to nonfull subcategories. For example, we find some conditions forcing a simple deflation functor to remain simple as a Mackey functor. For an inflation functor M defined on abelian groups over a field of characteristic zero, we also obtain a criterion for M to be semisimple, in terms of the images of inflation and induction maps on M.