The dihedral group as a group of automorphisms

Suppose that D=〈α,β〉 is a dihedral group generated by two involutions α and β. Let D act on a finite group G in such a manner that CG(αβ)=1. We show that if CG(α) and CG(β) are both nilpotent of class c, then G is nilpotent and the class of G is bounded solely in terms of c. If both CG(α) and CG(β) are of exponent dividing e, then the exponent of G is bounded solely in terms of e and |D|. Previously, results of this kind were known only for groups acted on by Frobenius groups of automorphisms.