  Let $G$ be a finite group. Let $\sigma =\{\sigma_{i} | i\in I\}$ be a
partition of the set of all primes $\Bbb{P}$ and $n$ an integer. We write
$\sigma (n) =\{\sigma_{i} |\sigma_{i}\cap \pi (n)\ne \emptyset \}$, $\sigma (G)
=\sigma (|G|)$. A set $ {\cal H}$ of subgroups of $G$ is said to be a complete
Hall $\sigma $-set of $G$ if every member of ${\cal H}\setminus \{1\}$ is a
Hall $\sigma_{i}$-subgroup of $G$ for some $\sigma_{i}$ and ${\cal H}$ contains
exact one Hall $\sigma_{i}$-subgroup of $G$ for every $\sigma_{i}\in \sigma
(G)$. A subgroup $A$ of $G$ is called: (i) a $\sigma$-Hall subgroup of $G$ if
$\sigma (|A|) \cap \sigma (|G:A|)=\emptyset$; (ii) ${\sigma}$-permutable in $G$
if $G$ possesses a complete Hall $\sigma$-set ${\cal H}$ such that
$AH^{x}=H^{x}A$ for all $H\in {\cal H}$ and all $x\in G$. 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 $|A|$ for each subgroup $A$ of $G$. Some known results are
generalized.
