The permutability of ${\sigma }_i$-sylowizers of some ${\sigma }_i$-subgroups in finite groups

Let $\sigma =\{{\sigma }_i\mathrm{\mid }i\in I\}$ be a partition of the set of all primes $\mathbb{P}$, $G$ a finite group and $\sigma (G)=\{{\sigma }_i\mathrm{\mid }{\sigma }_i\cap \pi (\mathrm{\mid }G\mathrm{\mid })\mathrm{\ne }\mathrm{\varnothing }\}$. A subgroup $S$ of a group $G$ is called a ${\sigma }_i$-sylowizer of a ${\sigma }_i$-subgroup $R$ in $G$ if $S$ is maximal in $G$ with respect to having $R$ as its Hall ${\sigma }_i$-subgroup. The main aim of this paper is to investigate the influence of ${\sigma }_i$-sylowizers on the structure of finite groups. We obtained some new characterizations of supersoluble groups by the permutability of the ${\sigma }_i$-sylowizers of some ${\sigma }_i$-subgroups.