On the orbits of plane automorphisms and their stabilizers

Let $\mathbb{k}$ be a perfect field with algebraic closure $\bar{\mathbb{k}}$. If $H$ is a subgroup of plane automorphisms over $\mathbb{k}$ and $p\in {\bar{\mathbb{k}}}^2$ 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 ${\bar{\mathbb{k}}}^2$. 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.