Let \(\mathcal X\) be a homomorph and let \(G\) be a finite group. A chief factor \(F\) of group \(G\) is an \(\mathcal X\)-chief factor of \(G\) if \(F\in{\mathcal X}\). It is denoted by \(C_{\mathcal X}(G)=\bigcap\{C_G(F)\mid F\) is an \(\mathcal X\)-chief factor of \(G\}\) if the set of \(\mathcal X\)-chief factors of \(G\) is nonempty, otherwise \(C_{\mathcal X}(G)=G\) and by \(R_{\mathcal X}(G)=\bigcap\{C_G(F)\mid F\) is a non-Abelian \(\mathcal X\)-chief factor of \(G\}\) if the set of non-Abelian \(\mathcal X\)-chief factors of \(G\) is nonempty, otherwise \(R_{\mathcal X}(G)=G\). It is clear that \(F(G)\leq C_{\mathcal X}(G)\leq R_{\mathcal X}(G)\). The author introduces two classes \(C({\mathcal X})=\{G\mid C_{\mathcal X}(G)=G\}\) and \(R({\mathcal X})=\{G\mid R_{\mathcal X}(G)=G\}\) which are \(N_0\)-closed formations. Moreover, if \(\mathcal X\) is \(N_0\)-closed, then both are Fitting classes. The author proves that the conditions \(C_G(C_{\mathcal X}(G))\leq C_{\mathcal X}(G)\) and \(F^*(G)\leq C_{\mathcal X}(G)\) are equivalent. The groups satisfying one of the above conditions are called \(C({\mathcal X})\)-constrained and as a consequence she obtains that if \(G\in R({\mathcal X})\) then \(C_G(C_{\mathcal X}(G))\leq C_{\mathcal X}(G)\). Finally, it is proved that if \(\mathcal X\) is \(N_0\)-closed, all groups such that \(G/G_{{\mathcal X}'}\) is \(C({\mathcal X})\)-constrained, have \(C({\mathcal X})\)-injectors, where \({\mathcal X}'\) is the Fitting class of the groups that have no section in \(\mathcal X\).