Profinite nature of Out(G) for all oligomorphic groups

Determine whether, for every oligomorphic permutation group G, the outer automorphism group Out(G) = Aut(G)/Inn(G) is profinite.

Background

The paper proves that for any oligomorphic group G, the group Out(G) is totally disconnected and locally compact, and that an open subgroup of Out(G) is profinite via a natural homomorphism from the normaliser N_G in Sym(ω) (Theorem 3.10). Moreover, Out(G) is shown to be profinite for two broad Borel classes of oligomorphic groups: those with no algebraicity and those with finitely many essential subgroups up to conjugacy (Theorem 4.7 and Theorem 1.3).

These results raise the broader structural question of whether Out(G) must be profinite in full generality, beyond the specific classes treated. The authors explicitly formulate this as a question following their upper bound result on Out(G).

References

Question 1.2. Is Out(G) always profinite for an oligomorphic group G?

— Oligomorphic groups, their automorphism groups, and the complexity of their isomorphism  (2410.02248 - Paolini et al., 2024) in Question 1.2, Section 1 (Introduction)

Let $M$ be an $\omega$-categorical structure in a first order finite language. Is the group $\operatorname{Out}(G)$ finite?

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in Section 2, subsection Oligomorphic groups

If $G$ is oligomorphic, is $\text{Out}(G)$ always profinite?

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in 2025 Section 6, subsection Profinite groups