On the generalized $\sigma$-Fitting subgroup of finite groups

Let \sigma =\{\sigma_{i} | i\in I\} be some partition of the set \mathbb P of all primes, and let G be a finite group. A chief factor H/K of G is said to be \sigma - central (in G ) if the semidirect product (H/K) \rtimes (G/C_{G}(H/K)) is a \sigma _{i} -group for some i=i(H/K) ; otherwise, it is called \sigma - eccentric (in G ). We say that G is: \sigma - nilpotent if every chief factor of G is \sigma -central; \sigma - quasinilpotent if for every \sigma -eccentric chief factor H/K of G , every automorphism of H/K induced by an element of G is inner. The product of all normal \sigma -nilpotent (respectively \sigma -quasinilpotent) subgroups of G is said to be the \sigma - Fitting subgroup (respectively the generalized \sigma - Fitting subgroup ) of G and we denote it by F_{\sigma}(G) (respectively by F^{*}_{\sigma}(G) ). Our main goal here is to study the relations between the subgroups F_{\sigma}(G) and F^{*}_{\sigma}(G) , and the influence of these two subgroups on the structure of G .