Galois points and Cremona transformations

In this article, we study Galois points of plane curves and the extension of the corresponding Galois group to $\mathrm{Bir}(\mathbb{P}^2)$ . We prove that if the Galois group has order at most $3$ , it always extends to a subgroup of the Jonquières group associated with the point $P$ . Conversely, with a degree of 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 $P$ . In addition, we also give an example of a Galois extension whose Galois group cannot be extended to Cremona transformations.