A subgroup $H$ of a group $G$ is called conjugately dense in $G$ if $H$ has nonempty intersection with each class of conjugate elements in $G$. We know that if $G$ is finite, then $H=G$. Now, my question is this:
Let $A\leq \operatorname{Aut}(G)$. Subgroup $H$ of a group $G$ is called $A$-conjugately dense in $G$ if $H$ has nonempty intersection with each $A$-orbit in $G$. If $G$ is a finite group, under what conditions can we say that $H=G$?
