The inverse Galois problem for Cherednik algebras

Given the spherical subalgebra $B$ of a rational Cherednik algebra, we aim to classify all finite groups $\Gamma$ for which there exists a domain $R$ on which $\Gamma$ acts by ring automorphisms, such that $B=R^{\Gamma}.$ We describe such groups in terms of geometry of the center of the reduction of $B$ modulo a large prime.