On some generalizations of permutability and $S$-permutability

Let $H$ and $X$ be subgroups of a finite group $G$. Then we say that $H$ is: $X$-quasipermutable (respectively, $X_S$-quasipermutable) in $G$ provided $G$ has a subgroup $B$ such that $G=N_G(H)B$ and $H$ $X$-permutes with $B$ and with all subgroups (respectively, with all Sylow subgroups) $V$ of $B$ such that $(|H|,|V|)=1$; $X$-propermutable (respectively, $X_S$-propermutable) in $G$ provided $G$ has a subgroup $B$ such that $G=N_G(H)B$ and $H$ $X$-permutes with $B$ and with all subgroups (respectively, with all Sylow subgroups) of $B$.
In this paper we analyze the influence of $X$-quasipermutable, $X_S$-quasipermutable, $X$-propermutable and $X_S$-propermutable subgroups on the structure of $G$.