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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.
}, issn = {2707-8523}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cmr/19078.html} }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.