Key subgroups in the Polish group of all automorphisms of the rational circle

Extending some results of a joint work with E. Glasner, we continue to study the Polish group $G:=\mathrm{A}\mathrm{u}\mathrm{t}({\mathbb{Q}}_0)$ of all circular order preserving permutations of the rational circle ${\mathbb{Q}}_0=\mathbb{Q}\mathrm{/}\mathbb{Z}$, endowed with the pointwise topology. We show that the point stabilizers $H=G_q$ are extremely amenable inj-key subgroups of $G$ (that is, they distinguish coarser Hausdorff group topologies on $G$), but are not co-minimal in $G$. These examples answer a question posed in a joint work with M. Shlossberg and are inspired by a question of V. Pestov concerning Polish groups with metrizable universal minimal flow. It remains an open problem to study Pestov’s question in its full generality.