  Let $R$ be a discrete valuation ring with field of fractions $K$ and perfect
residue field $k$ of characteristic $p>0$. Given a finite flat and commutative
group scheme $G$ over $K$ and a smooth projective curve $C$ over $K$ with a
rational point, we study the extension of pointed fppf $G$-torsors over $C$ to
pointed torsors over a regular model of $C$ over $R$. We first study this
problem in the category of log schemes: we prove that extending a $G$-torsor
into a log flat torsor amounts to finding a finite flat model of $G$ over $R$
for which a certain group scheme morphism to the Jacobian $J$ of $C$ extends to
the N\'eron model of $J$. Assuming this condition is satisfied, we give a
necessary and sufficient condition for the torsor to have an fppf-extension.
Finally, we give two detailed examples of extension of torsors when $C$ is a
hyperelliptic curve defined over $\mathbb{Q}$. In a second part, generalizing a
result of Chiodo, we give a criterion for the $r$-torsion subgroup of the
N\'eron model of $J$ to be a finite flat group scheme, and we combine it with
the results of the first part.
