Finite quotient of a manifold

Question

Suppose you have a real manifold M, and a group G acting freely on it. Suppose the quotient $M/G$ is finite. It is true that, as intuition suggest, the space $M/G$ is then discrete?

Edit: I added the "free" hypothesis thanks to the example of "Lord Shark the Unknown"

Answer 1

How about $M=\Bbb R$ and $G=\Bbb R^*$ acting on $M$ by multiplication. The quotient has two points, but I reckon it's not discrete.