= Solution
In $M$, take $T=\bigcup_{\alpha<\kappa}{}^\alpha2$ ordered by extension. Strong inaccessibility gives $|T_\alpha|=2^{|\alpha|}<\kappa$. In $M[H]$, each of these ground levels is therefore countable, while the height $\kappa$ is $\omega_1$. Thus the unchanged ground <set-theoretic tree> is an $\aleph_1$-tree in the extension.
Every ground binary <function> of length $\kappa$ still yields a distinct <cofinal branch> through this <set-theoretic tree>. There are $(2^\kappa)^M\ge(\kappa^+)^M$ such branches. The $\kappa$-chain condition preserves <cardinals> at and above $\kappa$, so $(\kappa^+)^M=\aleph_2^{M[H]}$. The ground branch family still has at least that <cardinality>. Consequently
$$
\boxed{M[H]\models\text{Kurepa's Hypothesis}.}
$$
This <Kurepa tree from an inaccessible binary tree> uses ground-model levels and branches; it does not claim the full binary <set-theoretic tree> newly computed in the extension has countable levels.
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