 Solution

Choose $h\in H\setminus \mathrm{Stab}(x)$ and put $y=hx\ne x$. For any $z\ne x$, double transitivity supplies an element $s\in G$ taking the ordered pair $(x,y)$ to $(x,z)$. Since $sx=x$, one hasThen $sh\in H$ sends $x$ to $z$. Together with the identity for $z=x$, this proves that $H$ acts transitively on $X$.

Now let $g\in G$. Transitivity of $H$ gives $h_0\in H$ with $h_{0}x=gx$. Henceso $g\in H$. ThereforeThis is the point-stabilizer maximality property of a doubly transitive group action.