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

Label the four vertices cyclically by $\mathbb{Z}/4$. Reading the arrows in the diagram, an $a$-edge moves one step in one direction and a $b$-edge one step in the other. Thus the monodromy homomorphism is(up to simultaneously reversing the chosen numbering). By the Galois correspondence for covering spaces, the subgroup associated with the chosen vertex above $x_0$ is