On $C$-$\mathcal{H}$-permutable subgroups of finite groups

Let $\sigma =\{\sigma_i \mid i\in I\}$ be some partition of the set of all primes ${\Bbb P}$, let $G$ be a finite group, and $\sigma(G)=\{\sigma_i\mid\sigma _i\cap \pi(G)\neq \emptyset\}$. A set $\mathcal{H}$ of subgroups of $G$ is a complete Hall $\sigma $-set of $G$ if every nonidentity member of $\mathcal{H}$ is a Hall $\sigma _i$-subgroup of $G$ for some $i\in I$ and $\mathcal{H}$ includes exactly one Hall $\sigma_ i$-subgroup of $G$ for every $\sigma_ i\in \sigma(G)$. Let $\mathcal{H}$ be a complete Hall $\sigma$-set of $G$ and let $C$ be a nonempty subset of $G$. We say that a subgroup $H$ of $G$ is $C$-$\mathcal{H}$-permutable if for all $A\in \mathcal{H}$ there exists some $x\in C$ such that $H^xA=AH^x$. We investigate the structure of $G$ by assuming that some subgroups of $G$ are $C$-$\mathcal{H}$-permutable. Some known results are generalized.