  Suppose that a finite group $G$ admits a Frobenius group of automorphisms FH
of coprime order with cyclic kernel F and complement H such that the fixed
point subgroup $C_G(H)$ of the complement is nilpotent of class $c$. It is
proved that $G$ has a nilpotent characteristic subgroup of index bounded in
terms of $c$, $|C_G(F)|$, and $|FH|$ whose nilpotency class is bounded in terms
of $c$ and $|H|$ only. This generalizes the previous theorem of the authors and
P. Shumyatsky, where for the case of $C_G(F)=1$ the whole group was proved to
be nilpotent of $(c,|H|)$-bounded class. Examples show that the condition of
$F$ being cyclic is essential. B. Hartley's theorem based on the classification
provides reduction to soluble groups. Then representation theory arguments are
used to bound the index of the Fitting subgroup. Lie ring methods are used for
nilpotent groups. A similar theorem on Lie rings with a metacyclic Frobenius
group of automorphisms $FH$ is also proved.
