= Solution
Here is the <double-coset criterion for a one-point extension>. Let $H=G_\alpha$, and suppose $x$ swaps $\omega$ and $\alpha$. Then $\langle G,x\rangle$ is a one-point extension if and only if
$$
\boxed{x^2\in H,\qquad xHx^{-1}=H,\qquad xgx\in GxG\ \text{for every }g\in G\setminus H.}
$$
For necessity, in an extension $H$ fixes both $\omega$ and $\alpha$, so $x$ normalizes it and $x^2$ lies in it. Since $G$ is transitive on $\Omega$, the extension has exactly two <double cosets> relative to $G$: $G$ and $GxG$. For $g\notin H$, $xgx$ moves $\omega$ into $\Omega$ and is in the latter <double coset>.
For sufficiency, the displayed conditions make $G\cup GxG$ closed under multiplication. Products with middle element in $H$ reduce using $xhx=(xhx^{-1})x^2\in G$; those with middle element outside $H$ remain in $GxG$. A finite nonempty multiplication-closed set of permutations containing the identity is a group. It contains $G$ and $x$, hence equals $\langle G,x\rangle$. Every element in $GxG$ moves $\omega$, while $G$ fixes it, giving the required <stabilizer subgroup>. The group is transitive because $G$ is transitive on $\Omega$ and $x$ moves the additional point.
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