Specify all $\alpha, \beta$ homomorphisms which make this sequence half-exact!
$$0 \to \mathbb{Z}\xrightarrow{\alpha} \mathbb{Z} \xrightarrow{\beta} \mathbb{Z}_2 \to 0$$
From the definition, we must fulfil $Im(\alpha) < Ker(\beta)$. $\alpha=id, \beta=$ modulo $2$ is a good solution, since we have:
$$Im(\alpha)=\mathbb{Z} \subseteq Ker (\beta)=2\mathbb{Z} \cong \mathbb{Z} $$
I have no idea if there are more solutions, or if not, how to prove that.
Any help appreciated.