Let $G$ be a group scheme over a scheme $S$ which is the spectrum of a discrete valuation ring. Let $\eta$ (resp. $s$) be the generic (resp. closed) point. Assume that the generic fiber $G_{\eta}$ is proper but $G_s^{0}$ is not and that $G_s$  has many connected components. Is there any contradiction/counterexample in finding a subscheme $H\subset G$ such that $H_{\eta}=G_{\eta}$, $G_s^0\subset H_s$ and $H_s$ proper (in particular, $H_s\subset G_s$)?