On $H_{\sigma }$-permutably embedded and $H_{\sigma }$-subnormaly embedded subgroups of finite groups

Let $G$ be a finite group. Let $\sigma =\{{\sigma }_i\mathrm{\mid }i\in I\}$ be a partition of the set of all primes $\mathbb{P}$ and $n$ an integer. We write $\sigma (n)=\{{\sigma }_i\mathrm{\mid }{\sigma }_i\cap \pi (n)\mathrm{\ne }\mathrm{\varnothing }\}$, $\sigma (G) =\sigma (|G|)$\mathrm{.}Aset$ {\cal H}$ofsubgroupsof$G$ is said to be a complete Hall $\sigma$-set of $G$ if every member of $\mathcal{H}\setminus \{1\}$ is a Hall ${\sigma }_i$-subgroup of $G$ for some ${\sigma }_i$ and $\mathcal{H}$ contains exact one Hall ${\sigma }_i$-subgroup of $G$ for every $\sigma_{i}\in \sigma (G)$\mathrm{.}Asubgroup$A$of$G$iscalled:(i)a$\sigma$-Hallsubgroupof$G$if$\sigma (|A|) \cap \sigma (|G:A|)=\emptyset$;(ii)${\sigma}$-permutablein$G$if$G$possessesacompleteHall$\sigma$-set${\cal H}$suchthat$AH^{x}=H^{x}A$forall$H\in {\cal H}$andall$x\in G$. We say that a subgroup $A$ of $G$ is $H_{\sigma }$-permutably embedded in $G$ if $A$ is a $\sigma$-Hall subgroup of some $\sigma$-permutable subgroup of $G$. We study finite groups $G$ having an $H_{\sigma }$-permutably embedded subgroup of order $\mathrm{\mid }A\mathrm{\mid }$ for each subgroup $A$ of $G$. Some known results are generalized.