 Solution

Letbe the group of finitely supported functions $f:G\to H$ with pointwise multiplication. The left-translation actiondefines the restricted wreath product

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 $H\wr G$.