$G$ transitive, some Stab$_G(x)$ transitive $\implies$ all Stab$_G(x)$ transitive

Question

I'm currently working through Dobson, Malnič, Marušič's Symmetry in Graphs and am stuck on the following problem (from Lemma 4.4.3):

Let $G$ be a (finite) group which acts transitively on a (finite) set $X$. If Stab$_G(x)$ is transitive on $X\backslash \{x\}$ for some $x\in X$, then Stab$_G(x)$ is transitive on $X\backslash \{x\}$ for all $x \in X$.

I am aware that $G$'s being transitive means that the stabilisers of the points are all conjugate in $G$, but am still unsure about how to proceed.

Any input would be appreciated.