Let $G$ be a group acting transitively on two sets $\Omega_{1}$ and $\Omega_{2}$. Also let $w_{i}\in\Omega_{i}$ and suppose there exists $\alpha\in Aut(G)$ such that $\alpha(G_{w_{1}})=G_{w_{2}}$, where $G_{w_{i}}$ denotes the point stabiliser of $w_{i}$. I would like to show that in this case the two actions of $G$ are equivalent, i.e. there exists a bijection $\phi: \Omega_{1} \to \Omega_{2}$ such that $$\phi(g(w_{1}))=\alpha(g)(\phi(w_{1})).$$
I'm not sure how to construct $\phi$ to start with.
I would really appreciate any help.