= Solution
Let $B=H_1\times\cdots\times H_m$, with each $H_i$ a copy of $H$. The <permutation> <group> $F\le S_m$ acts on $B$ by
$$
\alpha_\sigma(h_1,\ldots,h_m)_j=h_{\sigma^{-1}(j)}.
$$
Define the <permutation wreath product> to be the <semidirect product>
$$
\boxed{H\operatorname{wr}F=H^m\rtimes_\alpha F.}
$$
Explicitly,
$$
(b,\sigma)(c,\tau)=\bigl(b\,\alpha_\sigma(c),\sigma\tau\bigr).
$$
Writing $h_i$ for the element with entry $h$ in coordinate $i$ and identity elsewhere, and writing $\sigma$ for $(1,\sigma)$, <conjugation> satisfies
$$
\sigma h_i\sigma^{-1}=h_{\sigma(i)}.
$$
Thus the top <permutation> <group> permutes the coordinate subgroups in exactly its given action. Distinct coordinate subgroups commute, including when $H$ itself is nonabelian.
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