Some notes on the rank of a finite soluble group

Let $G$ be a finite group and let $\sigma=\{\sigma_i|i\in I\}$ be some partition of the set $\mathbb P$ of all primes. Then $G$ is called $\sigma$-nilpotent if $G=A_1\times\cdots\times A_r$, where $A_i$ is a $\sigma_{i_j}$-group for some $i_j=i_j(A_i)$. A collection $\mathscr H$ of subgroups of $G$ is a complete Hall $\sigma$-set in $G$ if each member $\ne1$ of $\mathscr H$ is a Hall $\sigma_i$-subgroup of $G$ for some $i\in I$ and $\mathscr H$ has exactly one Hall $\sigma_i$-subgroup of $G$ for every $i$ such that $\sigma_i\cap\pi(G)\ne\emptyset$. A subgroup $A$ of $G$ is called$\sigma$-quasinormal or $\sigma$-permutable [1] in $G$ if $G$ possesses a complete Hall $\sigma$-set $\mathscr H$ such that $AH^x=H^xA$ for all $H\in\mathscr H$ and $x\in G$. The symbol $r(G)$ ($r_p(G)$) denotes the rank ($p$-rank) $G$.
Assume that $\mathscr H$ is a complete Hall $\sigma$-set of $G$. We prove that (i) if $G$ is soluble, $r(H)\leq r\in\mathbb N$ for all $H\in\mathscr H$ and every $n$-maximal subgroup of $G$ $(n>1)$ is $\sigma$-quasinormal in $G$, then $r(G)\leq n+r-2$; (ii) if every member in $\mathscr H$ is soluble and every $n$-minimal subgroup of $G$ is $\sigma$-quasinormal in $G$, then $G$ is soluble and $r_p(G)\leq n+r_p(H)-1$ for all $H\in\mathscr H$ and odd $p\in\pi (H)$.