A finite group $G$ is called a generalized $PST$-group if every subgroup
contained in $F(G)$ permutes all Sylow subgroups of $G$, where $F(G)$ is the Fitting
subgroup of $G.$ The class of generalized $PST$-groups is not subgroup and quotient
group closed, and it properly contains the class of $PST$-groups. In this paper, the
structure of generalized $PST$-groups is first investigated. Then, with its help, groups
whose every subgroup (or every quotient group) is a generalized $PST$-group are determined, and it is shown that such groups are precisely $PST$-groups. As applications, $T$-groups and $PT$-groups are characterized.
