A $\sigma$-solubility criterion of a finite group

Throughout this paper, all groups are finite and $G$ always denotes a finite group. Moreover, $\sigma$ is some partition of the set of all primes $\mathbb{P}$, that is, $\sigma=\{\sigma_i\mid i\in I\}$, where $\mathbb{P}=\bigcup_{i\in I}\sigma_i$ and $\sigma_i\cap\sigma_j=\varnothing$ for all $i\ne j$. A set $\mathcal{H}$ of subgroups of $G$ is a complete Hall $\sigma$-set of $G$ if every member $\ne 1$ of $\mathcal{H}$ is a Hall $\sigma_i$-subgroup of $G$ for some $\sigma_i\in\sigma$ and $\mathcal{H}$ contains exactly one Hall $\sigma_i$-subgroup of $G$ for every $\sigma_i\in\sigma(G)$. A subgroup $A$ of $G$ is said to be: $\sigma_i$-permutable in $G$ if $G$ possesses a complete Hall $\sigma$-set $\mathcal{H}$ such that $AH^x=H^xA$ for all $H\in\mathcal{H}$ and all $x\in G$; $\sigma$-subnormal in $G$ if there is a subgroup chain $A=A_0\leqslant A_1\leqslant\dots\leqslant A_n=G$ such that either $A_{i-1}\unlhd A_i$ or $A_i/(A_{i-1})_{A_i}$ is $\sigma$-primary for all $i=1,\dots,n$. A subgroup $A$ of $G$ is said to be weakly $\sigma$-permutable in $G$ if there is a $\sigma$-permutable subgroup $S$ and a $\sigma$-subnormal subgroup $T$ of $G$ such that $G=AT$ and $A\cap T\leqslant S\leqslant A$. In this paper it is proved that if in every maximal chain $M_3<M_2<M_1<M_0=G$ of $G$ of length $3$ at least one of the subgroups $M_3$, $M_2$, or $M_1$ is either submodular or weakly $\sigma$-permutable in $G$, then $G$ is $\sigma$-soluble.