Galois action on ghost overlaps via centralizer quotient

Prove that for the field \(\GE\) generated by ghost overlaps and the ring class field \(H\), the Galois group \(\Gal(\GE/H)\) is canonically isomorphic to the quotient \(\mathcal{M}/\mathcal{S}\) (with \(\mathcal{M}\) a maximal abelian subgroup of \(\GL_2(\mathbb{Z}/\bar{d}\mathbb{Z})\) and \(\mathcal{S}\) the overlap symmetry group), and that under this isomorphism \(h(\tilde\mu_\p) = \tilde\mu_{F_h\p}\) holds for all \(h \in \Gal(\GE/H)\).

Background

The paper leverages an empirical isomorphism between a Galois group and a symmetry-quotient to organize and reconstruct ghost and SIC overlaps. Although numerically supported, the authors explicitly note they do not yet have a proof of this structural claim.