= Solution
A witness to failure of the <Axiom of power set> is $\boxed{\omega_1\in H_{\omega_2}}$. Every subset $A\subseteq\omega_1$ belongs to $H_{\omega_2}$, since its <transitive closure> has <cardinality> at most $\aleph_1$. If some $p\in H_{\omega_2}$ were the internal <power set> of $\omega_1$, transitivity and bounded subsethood would force $p$ to contain every ambient subset of $\omega_1$, and only those subsets. Thus it would be the full $\mathcal P(\omega_1)$.
By the <Cantor theorem>,
$$
|p|=2^{\aleph_1}\geq\aleph_2.
$$
Its <transitive closure> contains all its members, so $|\operatorname{tc}(\{p\})|\geq|p|\geq\aleph_2$. This contradicts $p\in H_{\omega_2}$. Therefore
$$
\boxed{H_{\omega_2}\not\models\mathrm{PowerSet}}.
$$
The <power-set failure in hereditarily small sets> needs no assumption about the <Continuum hypothesis>.
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