Let $A_K$ be an abelian variety defined over the fraction field $K$ of a d.v.r. $R$, and let $A/R$ be its Néron model. The special fiber $A_k$ of $A$ contains a canonical subgroup scheme $U$, the unipotent radical. If $U$ is zero, we say that $A_K$ has stable reduction over $R$. The starting point for this talk is a certain filtration of $A_k$ by closed unipotent subgroup schemes, introduced by B. Edixhoven. This talk will concern the case where the abelian variety is the Jacobian $J_K$ of a smooth curve $X/K$. We will discuss how one can obtain information about the filtration of $J_k$ by considering G-actions on $R'$-models of $X$, for certain tamely ramified G-Galois extensions $R'/R$. Furthermore, we will mention some numerical data for the filtration that we can compute with these methods, and how this relates to the stable reduction of X.