A functorial generalization of the Fitting subgroup in finite groups

Using the functional approach of R. Baer and B. I. Plotkin, we introduce and study the notion of $\mathcal{F}$-functorial whose values are characteristic subgroups of a finite group that possess certain properties of the Fitting subgroup. The lattice and semigroups of $\mathcal{F}$-functorials are described, the interrelation between $\mathcal{F}$-functorials and classes of groups is established, a characterization of their values is given in terms of group's elements inducing inner automorphisms on specified chief factors.