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The Fitting subgroup of a group $G$ has two generalizations; one of them is $F^*(G)$ and the other is‎ $\tilde{F‎‎}(G)‎‎$. The last one is defined by $\phi(G)\subset \tilde{F‎‎}(G)‎‎$ and‎ $\tilde{F‎‎}(G)‎‎/ \phi(G)=\operatorname{Soc}(G/ \phi(G))$ ‎‪‪[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 English resources correspond to this issue.

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A good reference is the book "Finite group Theory" of M. Aschbacher. Chapter 11 is devoted to the generalized Fitting subgroup $F^*(G)$ of $G$, and is quite detailed. The second one, $\tilde{F}(G)$ has been introduced by Peter Schmid (the reference is in German, I know, but it is discussed in Aschbacher's book and papers as well). Actually, I think that the paper of Murashka and Vasil’ev (which is in English) gives already good information on both generalized Fitting subgroups.

For further details see also here: Minimal normal subgroups of a finite group.

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