= Solution
Choose $h\in H\setminus\operatorname{Stab}(x)$ 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 has
$$
s\in\operatorname{Stab}(x)\subseteq H.
$$
Then $sh\in H$ 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 $h_0x=gx$. Hence
$$
h_0^{-1}g\in\operatorname{Stab}(x)\subseteq H,
$$
so $g\in H$. Therefore
$$
\boxed{H=G}.
$$
This is the point-stabilizer maximality property of a <doubly transitive group action>.
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