Moduli of finite flat torsors over nodal curves

We show that log flat torsors over a family $X\mathrm{/}S$ of nodal curves under a finite flat commutative group scheme $G\mathrm{/}S$ are classified by maps from the Cartier dual of $G$ to the log Jacobian of $X$. We deduce that fppf torsors on the smooth fibres of $X\mathrm{/}S$ can be extended to global log flat torsors under some regularity hypotheses.