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