  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.
