The Fitting subgroup of a group $G$ has two generalizations: the generalized Fitting subgroup $F^*(G)$ of Bender and $\tilde F(G)$ of Schmid. The latter is defined by $\tilde F(G)/\Phi(G) = \operatorname{Soc}(G/\Phi(G))$.
I saw these in V.I. Murashka, A.F. Vasil,ev, On partially conjugate-permutable subgroups of finite groups, but I do not have access to the sources presented in this paper.
I was wondering if you'd mind introducing me to English language resources on these subgroups.
Fitting subgroups are designed to be a non-identity ignorable subgroup of a finite group.
The generalized Fitting subgroup was introduced by Bender (1970).
The fundamental property is that $C_G( F^*(G) ) = Z(F^*(G)) = Z(F(G)) \leq F^*(G)$. For solvable groups, the corresponding result is $C_G( F(G) ) = Z(F(G)) \leq F(G)$, and $F(G)$ is minimal amongst all normal subgroups $N$ (of a solvable group) with $C_G(N) \leq N$.
$F^*(G)$ can be defined as the set of elements inducing inner automorphisms on chief factors, and $F(G)$ can be defined as the set of elements centralizing chief factors. In solvable groups, all chief factors are abelian, and so inner=central.
However, in solvable groups there is also a nice feature: the Fitting subgroup is the set of elements centralizing the complemented chief factors. In other words, one can ignore all Frattini factors, especially the Frattini subgroup. This latter version is often written as $F(G/\Phi(G)) = F(G)/\Phi(G)$.
In insoluble groups, this is no longer true. I have been studying the local version of this, as in Lafuente–Martínez-Pérez (2000). The global version is Schmid (1972)'s $\tilde F(G)$ (denoted by $M$). His subgroup is in general larger than $F^*(G)$ and simply forces the matter: $\tilde F(G)/\Phi(G) = F^*( G/\Phi(G) )$. Of course, $\tilde F(G/\Phi(G)) = \tilde F(G) / \Phi(G)$.
The view that the generalized Fitting subgroup is defined in terms of chief factors and quasinilpotent subgroups is in Huppert-Blackburn X.13. It also covers the traditional component+Fitting definition. Aschbacher 11.31 and Kurzweil–Stellmacher 6.5 only covers the traditional approach but are highly recommended. If you like the quasinilpotent approach, then see also example IX.2.5.c page 579-581 in Doerk–Hawkes.
Schmid's subgroup is also important, but I have found fewer sources on it. Aschbacher's textbook does not seem to discuss it. Ezquerro–Ballester-Bolinches uses it (denoted by $F'(G)$) in a critical way in Definition 1.4.9.
Both $F^*$ and a slight variation $\tilde F$ are discussed in I.1A.4 of Gorenstein-Lyons-Solomon. See 4.10 for the $\tilde F$ variation, and its very similar property. This entire section explains the point of the various Fitting subgroups.
