  Let $\sigma=\{\sigma_{i}|i\in I\}$ be a partition of the set of all primes
$\mathbb{P}$, $G$ a finite group and $\sigma(G)=\{\sigma_{i}|\sigma_{i}\cap
\pi(|G|)\neq\emptyset\}$. 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.
