Stabilizer as a hidden subgroup

For a group action $F:G\times X\to X$ and fixed $x$, the orbit map $f_x(g)=F(g,x)$ hides the stabilizer subgroup $G_x$: $f_x(g)=f_x(h)$ exactly when $h^{-1}g\in G_x$, equivalently when $g$ and $h$ lie in the same left coset of $G_x$.