Galois Points and Cremona Transformations

In this article, we study Galois points of plane curves and the extension of the corresponding Galois group to $\mathrm{B}\mathrm{i}\mathrm{r}({\mathbb{P}}^2)$. If the Galois group has order at most $3$, we prove that it always extends to a subgroup of the Jonquières group associated to the point $P$. In degree at least $4$, we prove that it is false. We provide an example of a Galois extension whose Galois group is extendable to Cremona transformations but not to a group of de Jonquières maps with respect to $P$. We also give an example of a Galois extension whose Galois group cannot be extended to Cremona transformations.