A group $G$ is said to be inner amenable if there exists a positive linear functional $m$ on $\ell^{\infty}(G)$ with $m(1)=1$ that is invariant under conjugation; i.e., $m(g^{-1}fg) = m(f)$ for every $f \in \ell^{\infty}(G)$, where $(g^{-1}fg)(x) = f(g^{-1}xg)$.
I have two related questions. First, if $(G_{n}, f_{ij})$ is a injective directed system of groups, does it follow that $\displaystyle \lim_{\rightarrow} G_n$ is inner amenable? Second, if $N$ is an inner amenable normal subgroup of $G$ such that $G/N$ is inner amenable, does it follow that $G$ is inner amenable?
I know that these questions can be answered affirmatively if all the groups are actually amenable. But I am not sure about the inner amenable case.
The first question has a positive answer.
Indeed, let $m_n$ be conjugacy invariant on $G_n$ with $m_n(\{e\})=0$, and push it forward to $\mu_n$ on the direct limit. Then $\mu_n(\{e\})=0$ using injectivity, and $\mu_n$ is $G_n$-invariant. Hence any cluster point of $(\mu_n)$ is a conjugacy-invariant mean as required.