  In this paper the concept of $\mathbb{F}$-functorial of a finite group was
introduced. These functorials have many properties of the Fitting subgroup of a
soluble group and the generalized Fitting subgroup of a finite group. It was
shown that the set of all $\mathbb{F}$-functorials is a complete distributive
lattice and the cardinality of this lattice is continuum. The sharp bounds on
the generalized Fitting height of mutually permutable product of two subgroups
were obtained.
