An $\mathfrak X$-crown of a finite soluble group

Let $G$ be a finite soluble group and $\Phi_\mathfrak X(G)$ an intersection of all those maximal subgroups $M$ of $G$ for which $G/\mathrm{Core}_G(M)\in\mathfrak X$. We look at properties of a section $F(G/\Phi_\mathfrak X(G))$, which is definable for any class $\mathfrak X$ of primitive groups and is called an $\mathfrak X$-crown of a group $G$. Of particular importance is the case where all groups in $\mathfrak X$ have equal socle length.