\b[No.] The <monodromy action of a covering space> for a two-sheeted covering takes values in $S_2$. Its nonidentity transposition commutes with every subgroup of $S_2$, so the <deck transformation group as a monodromy centralizer> always contains that transposition. Equivalently, every double cover has a <deck involution of a double covering> that exchanges the two points in each fibre. Thus a two-sheeted cover cannot have trivial deck group.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 21F / c / Solution

 Solution (source code)