On partially conjugate-permutable subgroups of finite groups

Let $R$ be a subgroup of a group $G$. We shall call a subgroup $H$ of $G$ the $R$-conjugate-permutable subgroup if $HH^r=H^rH$ for all $r\in R$. In this work the properties and the influence of $R$-conjugate-permutable subgroups (maximal, Sylow, cyclic primary) on the structure of finite groups are studied. As $R$ we consider the Fitting subgroup $F(G)$, quasinilpotent radical $F^*(G)$ and the generalized Fitting subgroup $\tilde{F}(G)$ that was introduced by P. Shmid. In particular, it was shown that group $G$ is nilpotent iff all its maximal subgroups are $\tilde{F}(G)$-conjugate-permutable.