The quotient manifold theorem says that

If $G$ is a Lie group acting freely and properly on a smooth manifold $M$ then $M/G$ has a (unique) smooth structure and the projection $\pi:M\to M/G$ is a submersion.

I was wondering what happens when the action is not free. My intuition suggests that we get corners, I have in mind this example: The action of $\frac{\mathbb{Z}}{2\mathbb{Z}}$ over $\mathbb{S}^2$ induced by the reflection wrt the $zy$-plane. The quotient manifold obtained is $\mathbb{D}^2$ and the boundary $\partial\mathbb{D}^2 $ can be identified with the fixed points of the action i.e. $\mathbb{S}^2\cap zy \text{-plane}$.

Does anyone know a theorem that covers the non-free case? Where can I read about it?

The quotient manifold theorem says that

If $G$ is a Lie group acting freely and properly on a smooth manifold $M$ then $M/G$ has a (unique) smooth structure such that the projection $\pi:M\to M/G$ is a submersion.

I was wondering what happens when the action is not free. My intuition suggests that we get corners, I have in mind this example: The action of $\frac{\mathbb{Z}}{2\mathbb{Z}}$ over $\mathbb{S}^2$ induced by the reflection wrt the $zy$-plane. The quotient manifold obtained is $\mathbb{D}^2$ and the boundary $\partial\mathbb{D}^2 $ can be identified with the fixed points of the action i.e. $\mathbb{S}^2\cap zy \text{-plane}$.

Does anyone know a theorem that covers the non-free case? Where can I read about it?