On subgroups that cover only $\mathfrak F$-central chief factors in finite groups

The authors call an element $x$ of a finite group $G$ $Q\mathfrak F$-supercentral if every chief factor $A/B$ of $G$ for which $x\in A\backslash B$ is $\mathfrak F$-central. The connection between $Q\mathfrak F$-supercentral elements of $G$ and its chief factors is investigated. In the case when $\mathfrak F$ is a nonempty saturated formation, the properties of subgroups that cover all $\mathfrak F$-central chief factors of $G$ and isolate all $\mathfrak F$-eccentric chief factors are investigated (the authors call these subgroups $\mathfrak F$-isolators). The connection between $\mathfrak F$-isolators and $\mathfrak F$-normalizers of $G$ is established.