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Suppose that a finite group G admits an automorphism 
of order 2n such that the fixed-point subgroup 
of the involution 
is nilpotent of class c. Let m = 
) be the number of fixed points of . 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.
