Morphisms of character varieties

Let $k$ be a field, let $H\subset G$ be (possibly disconnected) reductive groups over $k$, and let $\mathrm{\Gamma }$ be a finitely generated group. Vinberg and Martin have shown that the induced morphism of character varieties $${\underset{‾}{\mathrm{H}\mathrm{o}\mathrm{m}}}_{k\text{-gp}}(\mathrm{\Gamma },H)\mathrm{/}\mathrm{/}H\to {\underset{‾}{\mathrm{H}\mathrm{o}\mathrm{m}}}_{k\text{-gp}}(\mathrm{\Gamma },G)\mathrm{/}\mathrm{/}G$$ 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.