 Solution

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.