= Solution
Let
$$
B=\bigoplus_{g\in G}H_g
$$
be the group of finitely supported functions $f:G\to H$ with pointwise multiplication. The left-translation action
$$
(g\cdot f)(x)=f(g^{-1}x)
$$
defines the restricted <wreath product>
$$
\boxed{H\wr G=B\rtimes G.}
$$
Suppose $X$ and $Y$ are finite generating sets for $G$ and $H$. Embed each $y\in Y$ as a lamp supported at the identity of $G$. Conjugating these lamps by words in $X$ produces copies of $Y$ at every coordinate, and these copies generate $B$. Thus $X$ together with the identity-coordinate copy of $Y$ is a finite generating set for $H\wr G$.
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