  In this paper we study the groups all whose maximal or all Sylow subgroups
are $K$-$\mathfrak{F}$-subnormal in their product the with generalizations of
the Fitting subgroup $\mathrm{F}^*(G)$ and $\mathrm{\tilde F}(G)$. We prove
that a hereditary formation $\mathfrak{F}$ contains every group all whose Sylow
subgroups 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.
