  Suppose that a metacyclic Frobenius group $FH$, with kernel $F$ and
complement $H$, acts by automorphisms on a finite group $G$, in such a way that
$C_G(F)$ is trivial and $C_G(H)$ is nilpotent. It is known that $G$ is
nilpotent and its nilpotency class can be bounded in terms of $|H|$ and the
nilpotency class of $C_G(H)$. Until now, it was not clear whether the bound
could be made independent of the order of $H$. In this article, we construct a
family $\mathfrak{G}$ of finite nilpotent groups, of unbounded nilpotency
class. Each group in $\mathfrak{G}$ admits a metacyclic Frobenius group of
automorphisms such that the centralizer of the kernel is trivial and the
centralizer of the complement is abelian. This shows that the dependence of the
bound on the order of $H$ is essential.
