Towards Huppert–Shemetkov's theorem

It is proved that in every finite non-identity soluble group $G$ there exists a maximal subgroup $H$ such that $H$ does not contain the Fitting subgroup and $|G:H|=p^{r(G/\Phi(G))}$ for some prime number $p$. Here $r(G/\Phi(G))$ is the chief rank of the quotient $G/\Phi(G)$.