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 S_H,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 B_K which assigns to each group G in X its Burnside algebra B_K(G)=K_⊗ZB(G) where B(G) is the Burnside ring of G. We find the radical and the socle of B_K 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 B_K 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.