  Suppose that a finite group $G$ admits an automorphism $\varphi $ of order
$2^n$ such that the fixed-point subgroup $C_G(\varphi ^{2^{n-1}})$ of the
involution $\varphi ^{2^{n-1}}$ is nilpotent of class $c$. Let
$m=|C_G(\varphi)|$ be the number of fixed points of $\varphi$. It is proved
that $G$ has a characteristic soluble subgroup of derived length bounded in
terms of $n,c$ whose index is bounded in terms of $m,n,c$. A similar result is
also proved for Lie rings.
