Let $F$ be a finite flat group scheme over an algebraically closed field $k$ of characteristic 0. Note that I'm not assuming $F$ to be commutative.
Does the Cartier dual $F^D$ of $F$ exist? That is, is there a (finite) group scheme over $k$ representing the functor $S \mapsto Hom_S(F_S, \mathbb{G}_{m, S})$? [ I can't find any references in the literature, but I can't also see why $F^D$ shouldn't exist...]
If $F^D$ does indeed exist, I'm confused by the following: since $\mathbb{G}_m$ is commutative, $F^D$ and $(F^D)^D$ are both commutative. But shouldn't $(F^D)^D$ be isomorphic (non-canonically) to $F$? What if $F$ is not commutative?
Any help is appreciated.