I am trying to show that there exists a family of matrices $(M_n)$ in $GL(2, \mathbb{R}) $ such that $GL(2,\mathbb{Z}) M_n GL(2,\mathbb{Z})$ is the same for every n and $GL(2,\mathbb{Z}) M_n $ is different for every n.
I tried to simplify it by setting $$M_n = \begin{bmatrix}a_n & 0 \\ 0 & 1\end{bmatrix}$$
and find some suitable $a_n$, but so far nothing has worked.