Non-metacyclic groups acting with nilpotent centralizers

Let $A$ be a non-metacyclic finite group. Suppose that $A$ acts coprimely on a finite group $G$ in such a manner that $C_G(a)$ is nilpotent for any $a\in A^{\mathrm{\#}}$. In the present paper we investigate some conditions on $A$ which imply that $G$ is nilpotent with ``bounded’’ nilpotency class. More precisely, we generalize known results on action of $q$-groups and Frobenius groups, where $q$ is a prime.