  Suppose that a finite group $G$ admits a Frobenius group of automorphisms
$FH$ with kernel $F$ and complement $H$ such that the fixed-point subgroup of
$F$ is trivial: $C_G(F)=1$. In this situation various properties of $G$ are
shown to be close to the corresponding properties of $C_G(H)$. By using
Clifford's theorem it is proved that the order $|G|$ is bounded in terms of
$|H|$ and $|C_G(H)|$, the rank of $G$ is bounded in terms of $|H|$ and the rank
of $C_G(H)$, and that $G$ is nilpotent if $C_G(H)$ is nilpotent. Lie ring
methods are used for bounding the exponent and the nilpotency class of $G$ in
the case of metacyclic $FH$. The exponent of $G$ is bounded in terms of $|FH|$
and the exponent of $C_G(H)$ by using Lazard's Lie algebra associated with the
Jennings--Zassenhaus filtration and its connection with powerful subgroups. The
nilpotency class of $G$ is bounded in terms of $|H|$ and the nilpotency class
of $C_G(H)$ by considering Lie rings with a finite cyclic grading satisfying a
certain `selective nilpotency' condition. The latter technique also yields
similar results bounding the nilpotency class of Lie rings and algebras with a
metacyclic Frobenius group of automorphisms, with corollaries for connected Lie
groups and torsion-free locally nilpotent groups with such groups of
automorphisms. Examples show that such nilpotency results are no longer true
for non-metacyclic Frobenius groups of automorphisms.
