Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 21F / b / Solution

A covering map induces an injection on fundamental groups. There is no injection ${\mathbb{Z}}^2\hookrightarrow \mathbb{Z}/2$, and there is no injection $\mathbb{Z}/2\hookrightarrow {\mathbb{Z}}^2$ because ${\mathbb{Z}}^2$ is torsion-free. Thus neither proposed covering exists.