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For a formation $\mathfrak F$ , a subgroup M of a finite group G is said to be $\mathfrak F$ -pronormal in G if for each g ∈ G, there exists x ∈ 〈U,Ug〉 $\mathfrak F$ such that Ux = Ug. Let f be a subgroup embedding functor such that f(G) contains the set of normal subgroups of G and is contained in the set of Sylow-permutable subgroups of G for every finite group G. Given such an f, let fT denote the class of finite groups in which f(G) is the set of subnormal subgroups of G; this is the class of all finite groups G in which to be in f(G) is a transitive relation in G. A subgroup M of a finite group G is said to be $\mathfrak F$ -normal in G if G/CoreG(M) belongs to $\mathfrak F$ . A subgroup U of a finite group G is called K- $\mathfrak F$ -subnormal in G if either U = G or there exist subgroups U = U0 ≤ U1 ≤ . . . ≤ Un = G such that Ui–1 is either normal or $\mathfrak F$ -normal in Ui, for i = 1,2, …, n. We call a finite group G an $fT_{\mathfrak F}$ -group if every K- $\mathfrak F$ -subnormal subgroup of G is in f(G). In this paper, we analyse for certain formations $\mathfrak F$ the structure of $fT_{\mathfrak F}$ -groups. We pay special attention to the $\mathfrak F$ -pronormal subgroups in this analysis.