@DenisNardin a while back we were talking about the G-category $Fin_G$ of finite G-sets, and you suggested that at least when G is abelian, the $W(H)$-action on $(Fin_G)^H := (Fin^G)_{/(G/H)}$ should be trivial. is it possible that in fact $Fin_G$ is simply the unstraightening of the functor
$$ O_G \to P_G \to Grp \xrightarrow{B} Gpds \xrightarrow{Fun(-,Fin)} Cat^{op} ~ ? $$
here, $P_G$ is the poset of subgroups of $G$ as they're related by subconjugacy, and the special thing about G being abelian is that there's a functor from this to Grp (in fact to $Grp_{/G}$).