Extend maps between etale groups

Question

Let $V$ be a discrete valuation ring, $S=\operatorname{Spec}(V)$ and $\eta$ (resp. $s$) be the generic (resp. closed) point of $S$. Let $G$ and $H$ be flat group schemes over $S$ and assume I know that $G_{\eta}\cong H_{\eta}$ and $G_{s}\cong H_{s}$. Is it true that I have a true isomorphism $G\rightarrow H$? If not, is it true if we assume $G$ and $H$ etale over $S$? In my applications $G_s\cong \mathbb{Z}^r$ with $r$ a positive integer.

Answer 1

No. Consider $G$ the constant group $\mathbb Z\times S$ over $S$. Let $H$ be the subgroup of $G$ with $H_\eta=G_\eta$ and $H_s=2G_s$. Then $H_s\simeq G_s$ abstractly. But $H$ is not isomorphic to $G$ because there are points of $H_\eta$ (those points corresponding to odd integers) without specialization in $H_s$, while every point of $G_\eta$ has a specialization in $G_s$.