  Let $k$ be a field, $C\to \Spec k$ be a stable curve and let $G$ be a finite
group acting faithfully on the curve $C\to \Spec k$. In this article, we
compute the vector space $\Ext^1_G(\Omega_{C/k}, \O_C)$, the sheaf
$\Omega_{C/k}$ being the sheaf of relative differentials. This vector space is
naturally isomorphic to the set of first order $G$-equivariant deformations of
$C$. The computation we do here will be used in a future article.
