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 Néron 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 Q.In a second part, generalizing a result of Chiodo [9], we give a criterion for the r-torsion subgroup of the Néron model of J to be a finite flat group scheme, and we combine it with the results of the first part.