Let $S=Spec(R)$ where $R$ is a Henselian local ring with fraction field $K$. Let $G$ and $G'$ be finite, flat group schemes of odd order over $S$ with isomorphic generic fibers (over $Spec(K)$). Does this isomorphism extend to one over $S$? When $R=\mathbb{Z}_p$, the answer is yes following Fontaine's discussion in his 1975 paper 'Groupes finis commutatifs sur les vecteurs de Witt' (where he works over Witt vectors of a perfect field. Mazur uses this result to prove Theorem I.4 in his Eisenstein Ideal paper). What happens when $R$ is any general local ring (not necessarily Henselian)? Are there rings other than $\mathbb{Z}_p$ for which the answer to the above question is also a yes?

No. Here is a counterexample. Let $R=\mathbf{Z}_p[\zeta_p]$, where $p$ is a prime number and $\zeta_p$ is a primitive $p$th root of unity. Let $G=\mu_p=\mathrm{Spec}(R[x]/(x^p1))$ and let $G'$ be the constant group scheme $\mathbf{Z}/p\mathbf{Z}$. Then $G$ and $G'$ are not isomorphic because the special fiber of $G$ is connected but that of $G'$ isn't. But there is an isomorphism $G'\to G$ over the fraction field of $R$ given by $1\mapsto \zeta_p$. A lot more is known about these things, but not so much more by me. There are a few people around here who could probably say a lot more. 


The answer to the edited question is 'yes' if $R$ is a complete DVR in mixed characteristic with absolute ramification index $e< p1$ (where $p$ is the residue characteristic). This is one of the main results of Raynaud's (p,...,p) paper. He shows in Corollaire 3.3.6 that the functor sending a finite flat group scheme of $p$power order over R to its generic fiber is fully faithful (when $e< p1$). 

