On finite $ca$-$\mathfrak{F}$-groups

Let $\mathfrak{F}$ be a class of groups. A finite group $G$ is called a $ca$-$\mathfrak{F}$-group if its every abelian chief factor of $G$ is $\mathfrak{F}$-central and every nonabelian chief factor of $G$ is a simple group. It is established that the class of $ca$-$\mathfrak{F}$-groups forms a composite formation. The properties of the products of normal $ca$-$\mathfrak{F}$-subgroups of finite groups are investigated.