Is there a skyscraper group scheme?


Let $S$ be a DVR. Is there a group scheme $\mathcal{G}$ over $S$ which is generically {1} trivial i.e, identity group, but at the closed point some nontrivial group $G$?


For example: Let $C$ be a curve over $S$ whose generic fibre is smooth of genus $g$ with no automorphism but the closed fibre is a semistable curve(which has nontrivial automorphisms). Now consider the group of $S$-automorphisms of this curve $C$ over $S$? My question: Does this curve C over S has any non trivial S-automorphism?



"skyscraper group scheme"
Yes.


Let $\mathbb{A}^{1,2}$ be the affine line with a double origin. Consider the natural map $\mathbb{A}^{1,2}\to \mathbb{A}^1$. (To define this map, let $0\_1$ and $0\_2$ be the origins in $\mathbb{A}^{1,2}$. The above map sends any $x\neq 0\_1, 0\_2$ to $x$. It sends $0\_1$ and $0\_2$ to the origin in $\mathbb{A}^1$.)


The above morphism realizes $\mathbb{A}^{1,2}$ as a quasi-finite flat group scheme over $\mathbb{A}^1$. (It is not a separated group scheme over $\mathbb{A}^1$, of course.)


Note that the generic fibre of this group scheme is the trivial group. The fibre over the origin is the group $\mathbb{Z}/2\mathbb{Z}$.


To get to the situation you desire, let $\mathrm{Spec} \mathbb{C}[[t]]\to \mathbb{A}^{1}$ be a dominant map whose image contains the origin. Now base-change the group scheme $\mathbb{A}^{1,2}\to \mathbb{A}^1$ along this morphism to get a group scheme 
$$ G\to \mathrm{Spec} \mathbb{C}[[t]]$$ with trivial generic fibre and a non-trivial special fibre. (Here $G=\mathbb{A}^{1,2}\times\_{\mathbb{A}^1} \mathrm{Spec} \mathbb{C}[[t]]$.)


