On finite $ca$-$\mathfrak F$ groups and their applications
  Let $\mathfrak F$ be a class of groups. A group $G$ is called $ca$-$\mathfrak F$-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, we investigate the structure of a finite $ca$-$\mathfrak F$-group. Properties of mutually permutable products of finite $ca$-$\mathfrak F$-groups are studied. 
Evgeniy N. Myslovets, Alexander F. Vasil'ev
math.GR