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 $\mathcal{F}$ be a set of subgroups of $G$ which is closed under taking subgroups and taking conjugations by elements in $G$. Then let $\frak{F}$ denote the category whose objects are elements in $\mathcal{F}$ and whose morphisms are triples $(H, g, K)$ such that $H$, $K \in \mathcal{F}$ and $g \in G$ with $gHg^{-1} \subset K$. Taking the inverse limit of $A(H)$, where $H \in \mathcal{F}$, we obtain the ring $A(\frak{F})$ and the restriction homomorphism ${\rm{res}}^G_{\mathcal{F}} : A(G) \to A(\frak{F})$. We study this restriction homomorphism.