Let $\operatorname{\cdot}$ be a right group action of $G$ on a set $M$ Then

Proof:

$$ x \in G_{\alpha\cdot g} \leftrightarrow \alpha\cdot (gx) = \alpha \cdot g \leftrightarrow \alpha \cdot (gxg^{-1}) = \alpha \leftrightarrow gxg^{-1} \in G_{\alpha} \\ \leftrightarrow x \in g^{-1}\alpha g$$

and therefore $G_{\alpha\cdot g} =g^{-1}G_{\alpha}g$

Thus all stabilizer subgroups for elements of the orbit $G(\alpha)$ of $\alpha$ are conjugate to $G_{\alpha}$

Footnotes

... $g \in G$¹

$G_{\alpha}$ is the stabilizer subgroup of $\alpha \in M$