Consider the (Lamplighter) group $(\bigoplus_{n=-\infty}^{n=\infty}\mathbb{Z}_2) \rtimes_\phi\mathbb{Z}$, where $\phi(1)$ "shifts" every element in $\bigoplus_{-\infty}^{\infty}\mathbb{Z}_2$ to the right by $1$. I was wondering if its subgroup, $(\bigoplus_{n=-\infty}^{n=\infty}\mathbb{Z}_2) \rtimes_\phi2 \mathbb{Z}$ isomorphic to a wreath product of a finite abelian group with $\mathbb{Z}$.
I know that $\mathbb{Z}_2 \wr \mathbb{Z} \not \cong \mathbb{Z}_2 \wr 2 \mathbb{Z}$. However, $\mathbb{Z}_2 \wr 2 \mathbb{Z}$ is a wreath product of a finite abelian group with a group isomorphic to $\mathbb{Z}$, does that implies that it is isomorphic to a wreath product of a finite abelian group by $\mathbb{Z}$?