Suppose we have a non-singular rational matrix $Q$, consider the the $\mathbb{Z}$-span of the columns of $Q$ and $Q^{-1}$, denote it as $H = {\rm Span}_{\mathbb Z} \{ Q(\mathbb Z^{n}), Q^{-1}(\mathbb Z^{n})\} $. We know $H$ is isomorphic to $\mathbb Z^{n}$.
Question: can we describe the map from $\mathbb Z^{n} =\{ e_1,\ldots, e_n\}$ to $H$?
Thoughts: when $n=1$, and suppose $Q = pq^{-1}$, where $p$ and $q$ are comprime, then $${\rm Span}_{\mathbb Z} \{ Q(\mathbb Z), Q^{-1}(\mathbb Z) \} = \langle p^{-1} q^{-1} (e_1) \rangle.$$
Can we do something similar for higher dimensions? i.e. can we write $Q= MN^{-1}$, for some integer matrix $M$, $N$, such that $${\rm Span}_{\mathbb Z} \{ Q(\mathbb Z^{n}), Q^{-1}(\mathbb Z^{n})\} = \langle M^{-1} N^{-1}(e_i) \mid 1 \leq i \leq n\rangle?$$
In general, we can't just take $M = \frac{1}{q} I_n$, and $N = q Q$, where $q$ is the $\rm lcm$ of all the denominators of entries in $Q$.
Any hint or reference would be really appreciated.