On formations like $\mathfrak{CH}$

A description is given for formations of finite groups with a local screen $f$ such that for every group $A$ of automorphisms of an arbitrary group $G$ the fact that $A$ acts $f$-stably on the section $\widetilde{F}(G)/\Phi(G)$ implies that $A$ acts $f$-stably on $\Phi(G)$.