I'm really lost with this problem and I really need your help: Let $G=\mathbb{Z}^2\rtimes_A\mathbb{Z}$, and let $H\leq G$ with finite index in G. I have to prove that there is a subgroup $U$ of $\mathbb{Z}^2$ and $l\in\mathbb{Z}$ such that $H\simeq U\rtimes_{A^l}\mathbb{Z}$. Where $A=\left( \begin{array}{ccc} 2 & 1 \\ 1 & 1 \\ \end{array} \right)$ . The semiproduct in $G$ is defined: $(x_1,n_1)(x_2,n_2)=(x_1+A^{n_2}x_2,n_1+n_2)$.
Thanks for you hep