Is $G_\Delta$ acting transitively on $\Delta$? ($\Delta$ is an imprimitive block )

Is $G_\Delta$ acting transitively on $\Delta$? ($\Delta$ is an imprimitive
block )

Suppose $G$ acts non-primitively on set $X$ and let $\Delta$ be some block
of non-primitivity. It should be clear, that $G_\Delta$ is acting
transitively on $\Delta$, but I cannot see it.
Block stabilizer is defined: $G_\Delta=\{a\in G | \Delta^a=\Delta\}$.
I'm sure it will be a walk in the park for some of you. Thank you for your
answers.