Let $G$ be a group acting on a set $\Omega$. Then there exists a natural action of $G$ on $\Omega \times \Omega$ given by $(\alpha, \beta)^x = (\alpha^x, \beta^x)$. The orbits on $\Omega\times \Omega$ are called orbitals. For an orbital $\Delta$ and $\alpha \in \Omega$ set $$ \Delta(\alpha) := \{ \beta \in \Omega : (\alpha, \beta) \in \Delta \}. $$
If $G$ acts transitively on $\Omega$, and $\Delta(\alpha^x) = \Delta(\alpha^y)$, does this implies that $G_{\alpha} x = G_{\alpha} y$? (Of course by $G_{\alpha} x = G_{\alpha} y \Leftrightarrow \alpha^x = \alpha^y$ the reverse implication holds).