Is there a classification of involutions in $\text{GL}\_n(\mathbb{Z})$?


Here's some more details about what I mean. Consider $f \in \text{GL}\_n(\mathbb{Z})$ such that $f^2=1$. Regard $f$ as an automorphism of $\mathbb{Z}^n$. Extend $f$ to an automorphism $g$ of $\mathbb{Q}^n$. Then we can write $\mathbb{Q}^n = E\_1 \oplus E\_{-1}$, where $g$ acts as the identity on $E\_1$ and as $-1$ on $E\_{-1}$. Restricting this decomposition to $\mathbb{Z}^n$, we obtain a finite-index subgroup $A$ of $\mathbb{Z}^n$ and a decomposition $A = F\_1 \oplus F\_{-1}$ such that $f$ acts as the identity on $F\_1$ and as $-1$ on $F\_{-1}$.


However, we definitely cannot assume that $A = \mathbb{Z}^n$. For instance, the matrix whose first row is $(0 1)$ and whose second row is $(1 0)$ (by the way, I can't figure out how to get my matrices to display correctly) is an involution in $\text{GL}\_n(\mathbb{Z})$ that can be diagonalized over $\mathbb{Q}$ but not over $\mathbb{Z}$.


What else can be said here?



Involutions in GL_n(Z)
The problem is equivalent to classifying isomorphism classes of $n$-dimensional integral representations of the cyclic group $C\_2$ of order 2, or $\mathbb{Z}[C\_2]$-modules on $\mathbb{Z}^n$. This group has exactly 3 isomorphism classes of indecomposable free $\mathbb{Z}$-modules: 


(1) trivial 


(2) sign representation 


(3) 2-dimensional with matrix $\begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix}.$


Every $n$-dimensional $\mathbb{Z}[C\_2]$-module is a direct sum of (1), (2), (3) with uniquely determined multiplicities. Thus any involution is conjugate over $\mathbb{Z}$ to a block diagonal matrix with blocks [1], [-1], $\begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix}$ whose sizes are uniquely determined. 


