Rank and order of a finite group admitting a Frobenius group of automorphisms

Suppose that a finite group $G$ admits a Frobenius group $FH$ of automorphisms of coprime order with kernel $F$ and complement $H$. For the case where $G$ is a finite $p$-group such that $G=[G,F]$, it is proved that the order of $G$ is bounded above in terms of the order of $H$ and the order of the fixed-point subgroup $C_G(H)$ of the complement, while the rank of $G$ is bounded above in terms of $|H|$ and the rank of $C_G(H)$. Earlier, such results were known under the stronger assumption that the kernel $F$ acts on $G$ fixed-point-freely. As a corollary, for the case where $G$ is an arbitrary finite group with a Frobenius group $FH$ of automorphisms of coprime order with kernel $F$ and complement $H$, estimates are obtained which are of the form $|G|\le|C_G(F)|\cdot f(|H|,|C_G(H)|)$ for the order, and of the form $\mathbf r(G)\le\mathbf r(C_G(F))+g(|H|,\mathbf r(C_G(H)))$ for the rank, where $f$ and $g$ are some functions of two variables.