Realizing core quandles as twisted conjugation quandles

Determine, for an arbitrary group G, whether there exists a group H and an automorphism ψ ∈ Aut(H) such that the core quandle Core(G) is isomorphic, as a quandle, to the twisted conjugation quandle Conj(H, ψ) defined by the operation g ▷ h = ψ(h^{-1} g) h.

Background

This question arises from attempting to relate Bergman’s embedding of core quandles and Akita’s embedding of twisted conjugation quandles. In abelian groups, these structures coincide: Core(G) ≅ Conj(G, Inv) where Inv is inversion, as noted earlier in the paper.

For G = SU(2), it was verified that Conj(G, ψ) is not quandle-isomorphic to Core(G) for any ψ ∈ Aut(G), but it remains unclear whether Core(G) might be isomorphic to Conj(H, ψ) for some other group H with an automorphism ψ. The authors therefore formulate the general existence problem for arbitrary G.

References

However, we do not even know whether S{3}_{\mathbb{R}} is isomorphic, as a quandle, to any twisted conjugation quandle on a group. More generally, we pose the following problem. Let G be a group. Does there exist a group $H$ and a group automorphism \psi\in \operatorname{Aut}H such that \operatorname{Core} G is isomorphic, as a quandle, to \operatorname{Conj}(H,\psi)?

— An embedding of spherical quandles into Lie groups  (2603.29479 - Suzuki et al., 31 Mar 2026) in Section 7 (Results on the 3-spherical quandle S^3_ℝ), Question 7.1 (label: question_Core_twisted_conj)