Finite groups and Lie rings with an automorphism of order $2^n$

Suppose that a finite group $G$ admits an automorphism $φ$ of order $2^n$ such that the fixed-point subgroup $C_G(φ^{2^{n-1}})$ of the involution $φ^{2^{n-1}}$ is nilpotent of class $c$. Let $m=|C_G(φ)|$ be the number of fixed points of $φ$. 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.

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