Power-set failure in hereditarily small sets ($H_{{\omega }_2}/\models \mathrm{PowerSet}$)

The hereditarily small set structure $H_{{\omega }_2}$ fails the Axiom of power set at ${\omega }_1$. Every subset of ${\omega }_1$ belongs to $H_{{\omega }_2}$, but their full power set has cardinality at least ${\aleph }_2$ by the Cantor theorem, so cannot itself be hereditarily smaller than ${\aleph }_2$.