Finite groups admitting a dihedral group of automorphisms

Let $D=\langle \alpha, \beta \rangle$ be a dihedral group generated by the involutions $\alpha$ and $\beta$ and let $F=\langle \alpha \beta \rangle$. Suppose that $D$ acts on a finite group $G$ by automorphisms in such a way that $C_G(F)=1$. In the present paper we prove that the nilpotent length of the group $G$ is equal to the maximum of the nilpotent lengths of the subgroups $C_G(\alpha)$ and $C_G(\beta)$.