  Let $\Bbbk$ be a perfect field with algebraic closure $\overline{\Bbbk}$. If
$H$ is a subgroup of plane automorphisms over $\Bbbk$ and
$p\in\overline{\Bbbk}^2$ is a point, we describe the subgroup consisting of
plane automorphisms which stabilize the orbit of $p$ under $H$, when this orbit
has irreducible closure in $\overline{\Bbbk}^2$. As an application, we treat
the case where $H$ is cyclic and the closure of the orbit of $p$ is an
arbitrary (non-necessarily irreducible) curve.
