Solution
= Solution
Extend each element of $G$ to fix $\omega$. A <one-point extension of a permutation group> is a transitive permutation group $G^+\leq S_{\Omega^+}$ such that
$$
\boxed{G^+_\omega=G,}
$$
with the induced action on $\Omega$ equal to the prescribed action. The <stabilizer subgroup> equality is the substantive condition: merely adjoining an element that moves $\omega$ does not suffice. If $|\Omega|=n$, the <orbit-stabilizer theorem> gives $|G^+|=(n+1)|G|$.