Solution
= Solution
If $y=gx$, then
$$
\operatorname{Stab}_G(y)=g\operatorname{Stab}_G(x)g^{-1}.
$$
Indeed, $h$ fixes $x$ exactly when $ghg^{-1}$ fixes $gx$. Conjugation by $g$ is therefore the required isomorphism.
= Solution
If $y=gx$, then
$$
\operatorname{Stab}_G(y)=g\operatorname{Stab}_G(x)g^{-1}.
$$
Indeed, $h$ fixes $x$ exactly when $ghg^{-1}$ fixes $gx$. Conjugation by $g$ is therefore the required isomorphism.