There is a well-known result of Rosenblatt that if $G$ is non-amenable discrete group acting on the space $X$ and a stabilizer of each point $x\in X$ is amenable then the action of $G$ on itself is non-amenable. It is told that there is a following corollary: If $G$ is discrete group and H is its non-amenable subgroup. And for every $g \in G\setminus \{e\}$ the group $\{h\in G \mid hgh^{-1}=g\}$ is amenable, then $G$ is non-inner amenable.

Could you please explain me how does it follow from the theorem? I understand that it follows that the action of $H$ on itself is non-amenable, but not further.

Thanks in advance!