Outer automorphisms of oligomorphic groups in finite languages

Definitions and mathematical statement

Let $L$ be a finite first-order signature and let $M$ be a countably infinite $L$-structure. The structure is $\omega$-categorical if all countably infinite models of its complete theory are isomorphic to $M$.

Write $G=\operatorname{Aut}(M)$ and give $G$ the topology of pointwise convergence on the discrete underlying set of $M$. A basic neighborhood of $g\in G$ consists of all automorphisms agreeing with $g$ on a specified finite subset of $M$. Let $\operatorname{Aut}_{\mathrm{top}}(G)$ be the group of topological group automorphisms of $G$, and put \[ \operatorname{Inn}(G)=\{h\mapsto ghg^{-1}:g\in G\},\qquad \operatorname{Out}(G)=\operatorname{Aut}_{\mathrm{top}}(G)/\operatorname{Inn}(G). \]

For every such $\omega$-categorical structure $M$, is $\operatorname{Out}(\operatorname{Aut}(M))$ finite? The requested finiteness is for each structure separately; no uniform bound on its size is imposed.

Short English statement

Does the automorphism group of every countably infinite omega-categorical structure in a finite language have only finitely many outer topological automorphisms?

Sources

Andre Nies's section, Problem 2.1, in G. Barmpalias, S. Gao, J. He, T. Kihara, A. Nies, T. Slaman, C.-M. Tran, D. Turetsky, P. Welch, L. Yu and H. Zhang, Open Problems in Mathematical Logic, arXiv:2608.26628v2 (31 August 2026), https://arxiv.org/html/2608.26628v2#S2. The source lists the question as open; source checked on 24 October 2026.

Adapted from the cited source under CC BY 4.0, https://creativecommons.org/licenses/by/4.0/. The formulation and explanatory definitions have been adapted for this catalogue.