On the $\sigma_i$-length of a finite $\sigma$-soluble group

Let $\sigma=\{\sigma_i\mid i\in I\}$ be some partition of the set of all primes $\mathbb{P}$ and $G$ a finite group. $G$ is said to be $\sigma$-soluble if every chief factor $H/K$ of $G$ is a $\sigma_i$-group for some $i=i(H/K)$. We prove the following
Theorem. (i) If $G$ is $\pi$-separable, $H$ is a nilpotent Hall $\pi$-subgroup and $E$ a $\pi$-complement of $G$ such that $EX=XE$ for some subgroup $X$ of $H$ such that $H'\leqslant X\leqslant \Phi(H)$, then $l_\pi(G)\leqslant1$.
(ii) If $G$ is $\sigma$-soluble and $\{H_1,\dots, H_t\}$ is a Wielandt $\sigma$-basis of $G$ such that $H_i$ permutes with $H_j$ for all $i$, $j$, then $l_{\sigma_i}(G)\leqslant 1$ for all $i$.
(iii) If $G$ is $\sigma$-soluble and $\{H_1,\dots, H_t\}$ is a Wielandt $\sigma$-basis of $G$ such that $H_i$ permutes with $\Phi(H_j)$ for all $i$, $j$, then $l_{\sigma_i}(G)\leqslant 1$ for all $i$.
(iv) If $l_\pi(G)\leqslant 1$, then $QX=XQ$ each characteristic subgroup $X$ of $H$ and any Sylow subgroup $Q$ of $G$ such that $HQ=QH$.
(v) If $G$ is $\sigma$-soluble with $l_{\sigma_i}\leqslant 1$ for all $i$ and $\{H_1,\dots, H_t\}$ is a $\sigma$-basis of $G$, then each characteristic subgroup of $H_i$ permutes with each characteristic subgroup of $H_j$.