  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.
