Finite generalized $c$-supersoluble groups and their mutually permutable products

Let $\mathfrak{F}$ be a class of groups. A finite group $G$ is called a $ca$-$\mathfrak{F}$-group if its every non-abelian chief factor is simple and $H/K\leftthreetimes C_G(H/K)\in\mathfrak{F}$ for every abelian chief factor $H / K$ of $G$. In this paper the structure of finite $ca$-$\mathfrak{F}$-groups under the assumption that $\mathfrak{F}$ is a soluble saturated formation is found. Properties of mutually permutable products of finite $ca$-$\mathfrak{F}$-groups are studied.