Let $K$ be a field of characteristic $2$. Let $G = \mathbb Z / 2\mathbb Z$. The goal is to determine all indecomposable $KG$-modules up to isomorphism.
My main problem is to show that such a module is finite-dimensional.
What I did is to reformulate: Let $G=\langle g\rangle$, i.e. $g^2=1_G$. Then each $KG$-module can be described as a $K[T]$-module $V$ where the action of $T$ on $V$ can be described as $f\in\mathrm{End}_K(V)$ with $f^2=1_V$.
Note that $f$ satisfies the polynomial $T^2-1 = (T+1)^2$ (characteristic $2$), so the minimal polynomial $m(T)$ of $f$ is either $T+1$ or $T^2+1$. Also note that the only eigenvalue is $1$.
The first case that $m(T) = T+1$ is easy. Here we have $f=1_V$, so every subspace is also a submodule. Thus $V$ is indecomposable if and only if $(V,f)\cong(K,1)$. But for the second case, I don't know how to proceed.
It will be easier if $V$ is finite-dimensional. Then $f$ admits a Jordan normal form and one can proceed from there.
Since $(g-1)^2=0$ in $KG$, setting $x=g-1$ the group algebra $KG$ can be reinterpreted as $K[x]/(x^2)$, and in this guise the classification of its modules is addressed in this question.
Note that the answer that I posted there does not require the modules to be finite-dimensional.