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