Frobenius groups as groups of automorphisms

We show that if $GFH$ is a double Frobenius group with ``upper'' complement $H$ of order $q$ such that $C_G(H)$ is nilpotent of class $c$, then $G$ is nilpotent of $(c,q)$-bounded class. This solves a problem posed by Mazurov in the Kourovka Notebook. The proof is based on an analogous result on Lie rings: if a finite Frobenius group $FH$ with kernel $F$ of prime order and complement $H$ of order $q$ acts on a Lie ring $K$ in such a way that $C_K(F)=0$ and $C_K(H)$ is nilpotent of class $c$, then $K$ is nilpotent of $(c,q)$-bounded class.