Let $R$ be a complete DVR with fraction field $K$, characteristic $0$ and algebraically closed residue field $k$ of characteristic $p>0$. Suppose $G_{0}$ is a finite flat group scheme over $k$ so that we have the connected-étale sequence:
$0 \rightarrow (G_{0})^{0} \rightarrow G_{0} \rightarrow (G_{0})_{ét} \rightarrow 0$
Write $R_{n}=R/p^{n+1}$ and $G_{n}$ for the fibre of $G$ defined over $R_{n}$. I want to know how 'unique' finite flat group schemes $G_{1}$ are that are lifts of $G_{0}$ over $R_{1}$. In particular, such a scheme would have a connected-étale sequence:
$0 \rightarrow (G_{1})^{0} \rightarrow G_{1} \rightarrow (G_{1})_{ét} \rightarrow 0$
My understanding is that the étale group scheme $(G_{1})_{ét}$ is the unique lift of $(G_{0})_{ét}$. Perhaps by some duality/symmetry argument $(G_{1})^{0}$ is also unique in which case asking for a lift $G_{1}$ is the same as asking for an extension of group schemes (it would be interesting to know if this could be calculated).
(For context I am really interested in the case of an elliptic curve $E$ where $G$ is the $p$-torsion subgroup - maybe even for now assuming $E$ has ordinary reduction at $p$. In this case it is well known that if $G_{0}=E_{0}[p]$ is the $p$-torsion over $k$ then its connected-étale sequence is given by the split (connected-étale) exact sequence:
$0 \rightarrow \mu_{p} \rightarrow G_{0} \rightarrow \mathbb{Z}/p\mathbb{Z} \rightarrow 0$
I would like to know if I can calculate the possible (hopefully) finite number of lifts of $G_{0}$ explicitly.)