= Solution
Only the $k$th block can have nonzero subgradient coordinates. If $\beta^{(k)}\ne0$, the <Cauchy-Schwarz inequality> shows that the unique supporting vector is $\beta^{(k)}/\lVert\beta^{(k)}\rVert_2$. At zero, the defining inequality is $\lVert h\rVert_2\geq u^Th$ for every block vector $h$, which is equivalent to $\lVert u\rVert_2\leq1$. Thus
$$
\partial f_k(\beta)=
\begin{cases}
\{u:u^{(j)}=0\ (j\ne k),\ u^{(k)}=\beta^{(k)}/\lVert\beta^{(k)}\rVert_2\},&\beta^{(k)}\ne0,\\
\{u:u^{(j)}=0\ (j\ne k),\ \lVert u^{(k)}\rVert_2\leq1\},&\beta^{(k)}=0.
\end{cases}
$$
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