On weakly $\mathrm{tcc}$-subgroups of finite groups

The subgroups $A$ and $B$ are said to be {\sl $\mathrm{cc}$-permutable}, if $A$ is permutable with $B^x$ for some ${x\in \langle A,B\rangle}$. A subgroup $A$ of a finite group $G$ is called {\sl weakly $\mathrm{tcc}$-subgroup ($\mathrm{wtcc}$‑\hspace{0pt}subgroup, for brevity)} in $G$, if there exists a subgroup $Y$ of $G$ such that $G=AY$ and $A$ has a chief series ${1=A_0\leq A_1\leq \ldots \leq A_{s-1}\leq A_s=A}$ such that every $A_i$ is $\mathrm{cc}$-permutable with all subgroups of $Y$ for all $i=1, \ldots, s$. In this paper, we studied the influence of given systems of $\mathrm{wtcc}$-subgroups on the structure of a group $G$.