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 $C_N(B)=1$ then $C_{G/N}(A)=C_G(A)N/N$. If $N=G$ is a nilpotent group then we give as a corollary some description of the fixed points $C_{L(G)}(A)$ in the associated Lie ring $L(G)$ in terms of $C_G(A)$. 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 $A$).