  Let $k$ be a field, let $H \subset G$ be (possibly disconnected) reductive
groups over $k$, and let $\Gamma$ be a finitely generated group. Vinberg and
Martin have shown that the induced morphism of character varieties \[
\underline{\mathrm{Hom}}_{k\textrm{-gp}}(\Gamma, H)//H \to
\underline{\mathrm{Hom}}_{k\textrm{-gp}}(\Gamma, G)//G \] is finite. In this
note, we generalize this result (with a significantly different proof) by
replacing $k$ with an arbitrary locally noetherian scheme, answering a question
of Dat. Along the way, we use Bruhat-Tits theory to establish a few apparently
new results about integral models of reductive groups over discrete valuation
rings.
