First, let $A$ be a finitely generated *free* abelian group, and $s$ an automorphism of order $2$ of $A$. Set $G=\{1,s\}$. Then we know that $A$ is a sum of indecomposable $G$-lattices $A\_i$, where each $A\_i$ is either $(\mathbb{Z},1)$, or $(\mathbb{Z},-1)$, or $(\mathbb{Z}+\mathbb{Z},\sigma)$, where $\sigma$ interchanges the basis elements of 
$\mathbb{Z}+\mathbb{Z}$. Thus we know a "canonical form" of $(A,s)$.


Now let $A$ be a finite abelian $p$-group, where $p$ is a prime number (for example, $p=2$). Let $s$ be an automorphism of order $2$ of $A$. As before, set $G=\{1,s\}$. What is a "canonical form" of $(A,s)$? What are the indecomposable $G$-modules here?


If the question is too difficult, then at least what are interesting examples of indecomposable $G$-modules?



Involutive automorphisms of a finite abelian p-group
There is a big difference between the case $p=2$ and $p\neq 2$. 
If $p\neq 2$ then $A$ will be the direct sum of $(\mathbb{Z}/p^n,1)$ and $\mathbb{Z}/p^n,-1)$ for some $n$.
In this case all the indecomposables are also irreducible.


If $p=2$ and $A$ is 2-elementary abelian, then $A$ will split as the direct sum of $(\mathbb{Z}/2,1)$ and $(\mathbb{Z}\oplus\mathbb{Z},\sigma)$ where $\sigma$ interchanges the basis elements. In this case the first indecomposable is also irreducible, and the second one is not.


In case $A$ is a non elementary abelian 2-group, the situation is more complicated, and we can have many indecomposable modules. Here is how to construct such modules: Let $B=(b\_{i,j})$ be an $n\times n$ matrix over $\mathbb{Z}/2$. 
Let $A = \mathbb{Z}/4^n$ with basis $\{e\_i\}$ and let the action of $s$ be given by $s(e\_i) = e\_i + 2\sum\_j b\_{i,j}e\_j$
Then the module $A$ will be indecomposable if and only if the matrix $B$ is indecomposable (that is- it is not conjugate to any nontrivial block diagonal matrix).
Moreover, the conjugacy class of $B$ is an invariant of the isomorphism class of $A$. So non conjugate indecomposable matrices will give non isomorphic indecomposable modules. Since there are many such conjugacy classes, we get a variety of indecomposable modules.


