On weakly $m$-$\sigma$-permutably embedded subgroups of finite groups
Let $\sigma=\{{\sigma_i|i\in I}\}$ be some partition of the set of all primes $\mathbb{P}$ and $G$ a finite group. A group is said to be \textit{$\sigma$-primary} if it is a finite $\sigma_i$-group for some $i$.A subgroup $H$ of $G$ is said to be: \textit{$\sigma$-subnormal} in $G$ if there exists a subgroup chain $H=H_0\leq H_1\leq \cdots \leq H_n=G$ such that either $H_{i-1}$ is normal in $H_i$ or $H_i/(H_{i-1})_{H_i}$ is $\sigma$-primary for all $i=1,\ldots,n$; \textit{$\sigma$-permutably embedded} in $G$ if $H$ is $\sigma$-full and for every $\sigma_i\in \sigma(H)$, every Hall$\sigma_i$-subgroup of $H$ is also a Hall $\sigma_i$-subgroup of some $\sigma$-permutable subgroup of G.We say that a subgroup $H$ of $G$ is: \textit{$m$-$\sigma$-permutably embedded} in $G$ if$H=\langle A, B\rangle$ for some modular subgroup $A$ and $\sigma$-permutably embedded subgroup $B$ of $G$;\textit{weakly $m$-$\sigma$-permutably embedded} in $G$ if there are an $m$-$\sigma$-permutably embedded subgroup $L$ and a$\sigma$-subnormal subgroup $T$ of $G$ such that $G=HT$ and $H\cap T\leq L \leq H$.