Fixed points of the complements of Frobenius groups of automorphisms

Suppose that a finite group G admits a Frobenius group of automorphisms BA with kernel B and complement A. It is proved that if N is a BA-invariant normal subgroup of G such that (|N|, |B|) = 1 and CN(B) = 1 then CG/N(A) = CG(A)N/N. If N = G is a nilpotent group then we give as a corollary some description of the fixed points CL(G)(A) in the associated Lie ring L(G) in terms of CG(A). In particular, this applies to the case where GB is a Frobenius group as well (so that GBA is a 2-Frobenius group, with not necessarily coprime orders of G and A).