A survey article on some subgroup embeddings and local properties for soluble $\mathrm{PST}$-groups

Let $G$ be a group and $p$ a prime number. $G$ is said to be a $Y_p$-group if whenever $K$ is a $p$-subgroup of $G$ then every subgroup of $K$ is an $S$-permutable subgroup in $N_G(K)$. The group $G$ is a soluble $\mathrm{PST}$-group if and only if $G$ is a $Y_p$-group for all primes $p$.
One of our purposes here is to define a number of local properties related to $Y_p$ which lead to several new characterizations of soluble $\mathrm{PST}$-groups. Another purpose is to define several embedding subgroup properties which yield some new classes of soluble $\mathrm{PST}$-groups. Such properties include weakly S-permutable subgroup, weakly semipermutable subgroup, and weakly seminormal subgroup.