  Suppose that a finite $p$-group $P$ admits a Frobenius group of automorphisms
$FH$ with kernel $F$ that is a cyclic $p$-group and with complement $H$. It is
proved that if the fixed-point subgroup $C_P(H)$ of the complement is nilpotent
of class $c$, then $P$ has a characteristic subgroup of index bounded in terms
of $c$, $|C_P(F)|$, and $|F|$ whose nilpotency class is bounded in terms of $c$
and $|H|$ only. Examples show that the condition of $F$ being cyclic is
essential. The proof is based on a Lie ring method and a theorem of the authors
and P. Shumyatsky about Lie rings with a metacyclic Frobenius group of
automorphisms $FH$. It is also proved that $P$ has a characteristic subgroup of
$(|C_P(F)|, |F|)$-bounded index whose order and rank are bounded in terms of
$|H|$ and the order and rank of $C_P(H)$, respectively, and whose exponent is
bounded in terms of the exponent of $C_P(H)$.
