The inverse limit of the Burnside ring for a family of subgroups of a finite group

Let G be a finite nontrivial group and A(G) the Burnside ring of G. Let F be a set of subgroups of G which is closed under taking subgroups and taking conjugations by elements in G. Then let F denote the category whose objects are elements in F and whose morphisms are triples (H; g;K) such that H, K ∈ F and g ∈ G with gHg^-1 ⊂ K. Taking the inverse limit of A(H), where H ∈ F, we obtain the ring A(F) and the restriction homomorphism res_F ^G: A(G) → A(F). We study this restriction homomorphism.