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In this paper we study the groups all whose maximal or all Sylow subgroupsare $K$-$\mathfrak{F}$-subnormal in their product the with generalizations ofthe Fitting subgroup $\mathrm{F}^*(G)$ and $\mathrm{\tilde F}(G)$. We provethat a hereditary formation $\mathfrak{F}$ contains every group all whose Sylowsubgroups are $K$-$\mathfrak{F}$-subnormal in their product with$\mathrm{F}^*(G)$ if and only if $\mathfrak{F}$ is the class of all$\sigma$-nilpotent groups for some partition $\sigma$ of the set of all primes.We obtain a new characterization of the $\sigma$-nilpotent hypercenter, i.e.the $\mathfrak{F}$-hypercenter and the normal largest subgroup which$K$-$\mathfrak{F}$-subnormalize all Sylow subgroups coincide if and only if$\mathfrak{F}$ is the class of all $\sigma$-nilpotent groups.

On some applications of Fitting like subgroups of finite groups