On generalized $\sigma$-soluble groups

Let $\sigma =\{{\sigma }_i\mathrm{\mid }i\in I\}$ be a partition of the set of all primes $\mathbb{P}$ and $G$ a finite group. Let $\sigma (G)={\sigma _{i} : \sigma _{i}\cap \pi (G)\ne \emptyset$\mathrm{.}Aset${\cal H}$ofsubgroupsof$G$ is said to be a complete Hall $\sigma$-set of $G$ if every member $\mathrm{\ne }1$ of $\mathcal{H}$ is a Hall ${\sigma }_i$-subgroup of $G$ for some $i\in I$ and $\mathcal{H}$ contains exactly one Hall ${\sigma }_i$-subgroup of $G$ for every $i$ such that $\sigma _{i}\in \sigma (G)$\mathrm{.}Wesaythat$G$is$\sigma$-fullif$G$ possesses a complete Hall $\sigma$-set. A complete Hall $\sigma$-set $\mathcal{H}$ of $G$ is said to be a $\sigma$-basis of $G$ if every two subgroups $A,B\in \mathcal{H}$ are permutable, that is, $AB=BA$. In this paper, we study properties of finite groups having a $\sigma$-basis. In particular, we prove that if $G$ has a a $\sigma$-basis, then $G$ is generalized $\sigma$-soluble, that is, $G$ has a complete Hall $\sigma$-set and for every chief factor $H\mathrm{/}K$ of $G$ we have $\mathrm{\mid }\sigma (H\mathrm{/}K)\mathrm{\mid }\le 2$. Moreover, answering to Problem 8.28 in [A.N. Skiba, On some results in the theory of finite partially soluble groups, Commun. Math. Stat., 4(3) (2016), 281–309], we prove the following Theorem A. Suppose that $G$ is $\sigma$-full. Then every complete Hall $\sigma$-set of $G$ forms a $\sigma$-basis of $G$ if and only if $G$ is generalized $\sigma$-soluble and for the automorphism group $G\mathrm{/}{C}_G(H\mathrm{/}K)$, induced by $G$ on any its chief factor $H\mathrm{/}K$, we have either $\sigma (H\mathrm{/}K)=\sigma (G\mathrm{/}{C}_G(H\mathrm{/}K))$ or $\sigma (H/K) ={\sigma {i}}$and$G/C{G}(H/K)$isa$\sigma _{i} \cup \sigma _{j}$-groupforsome$i\ne j$.