On $\sigma$-semipermutable subgroups of finite groups

Let $\sigma =\{{\sigma }_i\mathrm{\mid }i\in I\}$ be some partition of the set of all primes $\mathbb{P}$, $G$ a finite group and $\sigma (G) ={\sigma_{i} |\sigma_{i}\cap \pi (G)\ne \emptyset }$\mathrm{.}Aset${\cal H}$ofsubgroupsof$Gis said to be a \emph{complete Hall\sigma -set} ofG$ifeverymember$\ne 1$of${\cal H}$isaHall$\sigma_{i}$-subgroupof$G$forsome$\sigma_{i}\in \sigma $and${\cal H}$containsexactoneHall$\sigma_{i}$-subgroupof$G$forevery$\sigma_{i}\in \sigma (G)$\mathrm{.}Asubgroup$H$of$G$ is said to be: \emph{$\sigma$-semipermutable in $G$ with respect to $\mathcal{H}$} if $H{H}_i^x=H_i^{x}H$ for all $x\in G$ and all $H_i\in \mathcal{H}$ such that $(\mathrm{\mid }H\mathrm{\mid },\mathrm{\mid }{H}_i\mathrm{\mid })=1$; \emph{$\sigma$-semipermutable in $G$} if $H$ is $\sigma$-semipermutable in $G$ with respect to some complete Hall $\sigma$-set of $G$. We study the structure of $G$ being based on the assumption that some subgroups of $G$ are $\sigma$-semipermutable in $G$.