Oligomorphic groups, their automorphism groups, and the complexity of their isomorphism

The paper follows two interconnected directions. 1. Let $G$ be a Roelcke precompact closed subgroup of the group $\text{Sym}(\omega )$ of permutations of the natural numbers. Then $\text{Inn}(G)$ is closed in $\text{Aut}(G)$, where $\text{Aut}(G)$ carries the topology of pointwise convergence for its (faithful) action on the cosets of open subgroups. Under the stronger hypothesis that~$G$ is oligomorphic, $N_G\mathrm{/}G$ is profinite, where $N_G$ denotes the normaliser of~$G$ in $\text{Sym}(\omega )$, and the topological group $\text{Out}(G)=\text{Aut}(G)\mathrm{/}\text{Inn}(G)$ is totally disconnected, locally compact. 2a. We provide a general method to show smoothness of the isomorphism relation for appropriate Borel classes of oligomorphic groups. We apply it to two such classes: the oligomorphic groups with no algebraicity, and the oligomorphic groups with finitely many {essential} subgroups up to conjugacy. 2b. Using this method we also show that if $G$ is in such a Borel class, then $\text{Aut}(G)$ is topologically isomorphic to an oligomorphic group, and $\text{Out}(G)$ is profinite.