Some new characterizations of $PST$-groups

Let $H$ and $B$ be subgroups of a finite group $G$ such that $G=N_G(H)B$. Then we say that $H$ is \emph{quasipermutable} (respectively \emph{$S$-quasipermutable}) in $G$ provided $H$ permutes with $B$ and with every subgroup (respectively with every Sylow subgroup) $A$ of $B$ such that $(\mathrm{\mid }H\mathrm{\mid },\mathrm{\mid }A\mathrm{\mid })=1$. In this paper we analyze the influence of $S$-quasipermutable and quasipermutable subgroups on the structure of $G$. As an application, we give new characterizations of soluble $PST$-groups.