On some criteria of $f$-hypercentral actions of finite groups

Let $G$ be an $A$-group and let $f$ be a formation function. Note that $A$ is said to act $f$-hypercentrally on $G$ if $A/C_A(H/K)\in f(p)$ on every $A$-composition factor $H/K$ of $G$ for all $ p\in \pi (H/K)$. We obtain some criteria of the actions to be $f$-hypercentral and generalize a few available results.