On finite $ca$-$\mathfrak{F}$ groups and their applications

Let $\mathfrak{F}$ be a class of groups. A group $G$ is called $ca$-$\mathfrak F$-groupifitseverynon-abelianchieffactorissimpleand$H/K \leftthreetimes C_G(H/K) \in \mathfrak F$foreveryabelianchieffactor$H/K$of$G$\mathrm{.}Inthispaper,weinvestigatethestructureofafinite$ca$-$\mathfrak F$-group\mathrm{.}Propertiesofmutuallypermutableproductsoffinite$ca$-$\mathfrak F$-groups are studied.