On weakly $\sigma$-quasinormal subgroups of finite groups

Let $\sigma=\{\sigma_{i}| i\in I\}$ be a partition of the set of all primes $\mathbb{P}$, and $G$ be a finite group. A set $\mathcal{H}$ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if every member $\ne 1$ of $\mathcal{H}$ is a Hall $\sigma_{i}$-subgroup of $G$ for some $i\in I$, and $\mathcal{H}$ contains exactly one Hall $\sigma_{i}$-subgroup of $G$ for every $i$ such that $\sigma_{i}\cap\pi(G)\ne\emptyset$. A group is said to be $\sigma$-primary if it is a finite $\sigma_{i}$-group for some $i$.
A subgroup $A$ of $G$ is said to be: $\sigma$-quasinormal in $G$ if $G$ possesses a complete Hall $\sigma$-set $\mathcal{H}$ such that $AH^{x}=H^{x}A$ for all $H\in\mathcal{H}$ and all $x\in G$; $\sigma$-subnormal in $G$ if there is a subgroup chain $A=A_{0}\leq A_{1}\leq\cdots\leq A_{t}=G$ such that either $A_{i-1}\trianglelefteq A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is $\sigma$-primary for all $i=1,\ldots,t$; weakly $\sigma$-quasinormal in $G$ if there are a $\sigma$-quasinormal subgroup $S$ and a $\sigma$-subnormal subgroup $T$ of $G$ such that $G=AT$ and $A\cap T\leq S\leq A$.
We study $G$, assuming that some subgroups of $G$ are weakly $\sigma$-quasinormal in $G$.