Finite groups with a solvable Hall $\sigma$-basis

Let $\sigma = \{\sigma_i \mid i \in I \}$ be a partition of the set $\mathbb{P}$ of all primes, and let $G$ be a finite group. A set $\mathcal {H}$ of subgroups of $G$ is called a complete Hall $\sigma$-set of $G$ if every subgroup in $\mathcal {H}$ is a $\sigma_i$-Hall subgroup of $G$ for every $i \in I$ and $\mathcal {H}$ contains exactly one $\sigma_i$-Hall subgroup for every $i$ such that $\sigma_i \cap \pi (G) \neq \varnothing$. In this paper, we study the structure of the group $G \in \bigcap_{i \in I}D_{\sigma_i}(\mathfrak {S})$ under the condition that all subgroups in every complete Hall $\sigma$-set of the group $G$ are permutable.