Inner amenability

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.

• If $G$ acts amenably on $X$ (in the sense of Eymard: there's an invariant mean) with amenable stabilizers then $G$ is amenable. So your statement is probably that if $G$ is non-amenable and acts on $X$ with amenable stabilizers then the action on $X$ (not on itself) is non-amenable. Now apply this to the action by conjugation of $G$ on $G\smallsetminus\{1\}$. – YCor Oct 8 '16 at 16:44
• For non-experts: $G$ inner amenable means that $G$ acts amenably (=preserves a mean defined on all subsets) by conjugation on $G\smallsetminus\{1\}$. Trivial examples are those $G$ with a nontrivial finite conjugacy class (=non-icc groups), and amenable groups $G$. – YCor Oct 8 '16 at 16:47