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Orbits of an action of $G$ on a set $X$

Let $G$ be a finite group of order $p^\alpha q^\beta$ where $p, q$ are distinct primes and $\alpha,\beta>0$. Let $X$ be a set of size $p^\alpha$ and $G$ acts on $X$ via the action map $G\times X\to X,\, (g,x)\mapsto g.x$.

Prove that there exists a subset $Y$ of $X$ such that $|\mathcal{O}_G(Y)|=q^\beta$.

$\textbf{My attempt:}$
Since the group $G$ is not cyclic, it is not necessarily true that a non-trivial element always has exactly a single orbit. Hence, $|\mathcal{O}_G(Y)|>1$ for any $Y\subseteq X$. My idea is to prove that $|\mathcal{O}_G(Y)|$ is independent of $Y\subseteq X$ as long as $Y$ is chosen such that $|Y|\geq q^\beta.$
Clearly, $Y
eq \emptyset$. Now, a $(g_1,g_2)\in G\times G$ maps $Y\times Y$ to
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