For a countable discrete group $G$ acting by homeomorphisms on a compact metrizable space $X$, we say that $G\curvearrowright X$ is (topologically) amenable if there exists a sequence of continuous maps $(\mu_n: X\to \mathrm{Prob}(G))_{n\in\mathbb{N}}$ (where $\mathrm{Prob}(G)$ is equipped with the $\ell^1$-norm) such that $$ \mathrm{sup}_{x\in X}\left\|\mu_n(g\cdot x)-g\cdot \mu_n(x)\right\|\xrightarrow{n\to\infty} 0, \quad \forall g\in G. $$
Denote by $\mathbb{F}_2$ the free group on two generators. Does there exist a countable discrete group $H$, and commuting actions of $\mathbb{F}_2$ and $H$ on a compact metrizable space $X$, so that the action of $\mathbb{F}_2$ is amenable and the action of $H$ is minimal?
For example, when $\mathbb{F}_2$ is acting on its Gromov boundary (or any other amenable $\mathbb{F}_2$-boundary), then this is not possible, because the only homeomorphism which commutes with the boundary action is the identity.
