Let M be an n dimensional topological manifold, may be non-compact. Suppose there is an action of a group G on M, the orbits are closed, but may not be bounded. Consider the projection $\pi: M \to M/G$, do we have path lifting and homotopy lifting property? If not, what other conditions should we add?

I know when M is a smooth manifold, and G acts smoothly, freely, and properly, then $\pi$ is a smooth normal covering. But we consider manifold which may not be smooth and we only need lifting properties, the condition is too strict.

Consider the example $\pi: \mathbb{R}^3\to \mathbb{R}^3/Z_2$, $\mathbb{R}^3/Z_2$ is the cone over $\mathbb{R}P^2$, not a topological manifold. In this example, do we have path lifting and homotopy lifting property?