Comparing the face rings of a boolean complex and its barycentric subdivision

We consider the relationship between the Stanley-Reisner ring (a.k.a. face ring) of a simplicial or boolean complex $\Delta$ and that of its barycentric subdivision. These rings share a distinguished parameter subring. S. Murai asked if they are isomorphic, equivariantly with respect to the automorphism group $\operatorname{Aut}(\Delta)$, as modules over this parameter subring. We show that, in general, the answer is no, but for Cohen-Macaulay complexes in characteristic coprime to $|\operatorname{Aut}(\Delta)|$, it is yes, and we give an explicit construction of an isomorphism. To give this construction, we adapt and generalize a pair of tools introduced by A. Garsia in 1980. The first one transfers bases from a Stanley-Reisner ring to closely related rings of which it is a Gr\"obner degeneration, and the second identifies bases to transfer.