Extensions of several coprime results to good action case

Let $G$ and $A$ be groups where $A$ acts on $G$ by automorphisms. We say "\textit{the action of $A$ on $G$ is good}" if the equality $$ holds for any subgroup $B$ of $A$ and for any $B$-invariant subgroup $H$ of $G$. It is straightforward that every coprime action is a good action. In the present work we extend some results due to Ward, Gross, Shumyatsky, Jabara, and Meng and Guo under coprime action to good action.