= Solution
Choose $1\ne h\in H$ and let $t$ generate $\mathbb Z$. For each binary string $(\varepsilon_0,\ldots,\varepsilon_{n-1})$, the word
$$
h^{\varepsilon_0}t h^{\varepsilon_1}t\cdots t h^{\varepsilon_{n-1}}t^{-(n-1)}
$$
records that string in the lamps at positions $0,1,\ldots,n-1$. The resulting $2^n$ group elements are distinct and have word length at most $3n$ with respect to any finite generating set containing $h$ and $t$. Hence the growth function is bounded below exponentially, and $H\wr\mathbb Z$ has exponential growth.
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