Maximality of connected components of finite flat group schemes Let $k$ be a perfect field of characteristic $p>0$. Let $K$ be a finite, totally ramified extension of $K_0:=\mathrm{Frac}\ W(k)$ and let $\mathcal{O}_K$ be the ring of integers of $K$. All group schemes in this question will be assumed to be commutative.
If $\mathscr{V}$ is a finite flat group scheme over $\mathcal{O}_K$, we say that $\mathscr{V}$ is $\mathit{maximal}$ if for any other finite flat (commutative) group scheme $\mathscr{W}$ over $\mathcal{O}_K$ which has the same generic fibre as $\mathscr{V}$, there is a morphism $\mathscr{V}\to\mathscr{W}$ which restricts to the identity on the generic fibre. By a theorem of Raynaud, we can replace any finite flat group scheme over $\mathcal{O}_K$ by a maximal one without changing the generic fibre. 
Now the question: Let $\mathscr{V}$ be a finite flat group scheme as before and suppose that $\mathscr{V}$ is maximal. Is it true that the connected component $\mathscr{V}^0$ of $\mathscr{V}$ is maximal as well?
 A: Yes, and the same holds with $\mathscr{V}^0$ replaced by any finite flat $O_K$-subgroup $\mathscr{V}'$ of $\mathscr{V}$ (and with $O_K$ replaced by any discrete valuation ring of generic characteristic 0).  Consider a map $f:\mathscr{V}' \rightarrow \mathscr{H}$ between finite flat $O_K$-group schemes that is an isomorphism on generic fibers.  We want to show that $f$ is an isomorphism.  
Form the pushout of the exact sequence 
$$1 \rightarrow \mathscr{V}' \rightarrow \mathscr{V} \rightarrow \mathscr{V}'' \rightarrow 1$$
against $f$ to get a short exact sequence of finite flat $O_K$-group schemes
$$1 \rightarrow \mathscr{H} \rightarrow \mathscr{G} 
\rightarrow \mathscr{V}'' \rightarrow 1.$$
There is an evident commutative diagram from the first of these to the second inducing $f$ along the left and the identity map for $\mathscr{V}''$ on the right, so the map in the middle $\mathscr{V} \rightarrow \mathscr{G}$ is an isomorphism on generic fibers (as the outer maps have that property).  By maximality of $\mathscr{V}$ it follows that this middle map is an isomorphism, so the same then holds for $f$ by the snake lemma or whatever for fppf abelian sheaves.
