I'm trying to solve the following exercise in Lee's book.
Suppose M is a connected, oriented smooth manifold and Γ is a discrete group acting freely and properly on M. We say the action is orientation-preserving if for each γ ∈ Γ, the diﬀeomorphism $x\rightarrow γx$ is orientation-preserving. Show that M/Γ is orientable if and only if Γ is orientation-preserving.
Assuming $\Gamma $ to be orientation-preserving, I tried taking a commutative diagram taking an orientation of M to one of M$/\Gamma$ and vice-versa and tried using local diffeomorphism arguments to prove the orientability of M$/\Gamma$. However, I can't make this proof explicit. Nor can I proceed for the other direction.
Can someone please provide a proof for both if and only if parts? Thanks in advance.