Duals of finite quantum permutation groups

We work out axioms for the duals $G\subset U_N^+$ of the finite quantum permutation groups, $F\subset S_N^+$ with $|F|<\infty$, and we discuss how the basic theory of such quantum permutation groups partly simplifies in the dual setting. We discuss as well some potential extensions to the infinite case, in connection with the well-known question of axiomatizing the discrete quantum group actions on the infinite graphs.