  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
$(|H|, |A|)=1$. 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.
