Homotopy theory of dihedral and quaternionic sets

Call a map f : A→B of O(2)-spaces a weak equivalence if f H :AH→BH is a weak homotopy equivalence for all finite cyclic and dihedral subgroups H⊆O(2). We show that the resulting homotopy category is equivalent to the homotopy category of dihedral sets with suitable weak equivalences. There is a similar relation between Pin(2)-spaces and quaternionic sets.