= 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
$$
\rho:F(a,b)\longrightarrow\mathbb Z/4,
\qquad
\rho(a)=-1,quad \rho(b)=1
$$
(up to simultaneously reversing the chosen numbering). By the <classification of connected covering spaces>[Galois correspondence for covering spaces], the subgroup associated with the chosen vertex above $x_0$ is
$$
\boxed{\ker\rho=\{w\in F(a,b):\#_b(w)-\#_a(w)\equiv0\pmod4\}}.
$$
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