Covering groups and their integral models

Given a reductive group $\boldsymbol{\mathrm{G}}$ over a base scheme $S$, Brylinski and Deligne studied the central extensions of a reductive group $\boldsymbol{\mathrm{G}}$ by $\boldsymbol{\mathrm{K}}_2$, viewing both as sheaves of groups on the big Zariski site over $S$. Their work classified these extensions by three invariants, for $S$ the spectrum of a field. We expand upon their work to study "integral models" of such central extensions, obtaining similar results for $S$ the spectrum of a sufficiently nice ring, e.g., a DVR with finite residue field or a DVR containing a field. Milder results are obtained for $S$ the spectrum of a Dedekind domain, often conditional on Gersten's conjecture.