Finite group actions, rational fixed points and weak N\'eron models

If $G$ is a finite $\mathrm{\ell }$-group acting on an affine space ${\mathbb{A}}^n$ over a finite field $K$ of cardinality prime to $\mathrm{\ell }$, Serre has shown that there exists a rational fixed point. We generalize this to the case where $K$ is a henselian discretely valued field of characteristic zero with algebraically closed residue field and with residue characteristic different from $\mathrm{\ell }$. We also treat the case where the residue field is finite of cardinality $q$ such that $\mathrm{\ell }$ divides $q-1$. To this aim, we study group actions on weak N'eron models.