Under what conditions is a quotient of a topological spaces homeomorphic to the original space?

Question

Let´s say, we have a topological space $X$ and its subspace $S$. Take the quotient space $X/S$.

When can we say that $X/S$ is homeomorphic to $X$?

When can we say that, given $X$ is also Hausdorff?

Thank you.

(This question is not from any paper or textbook, it just came to my mind.)