Let $M$ be a submanifold of the Euclidean space $\mathbb{R}^n$. Let $G$ be a finite group acting on $M$ freely. I want to compute the homology (or even the cohomology ring) of $M/G$.

Suppose the coordinates of all points in $M$ are given by some polynomial equations (or inequalities) of the $n$-variables of coordinates of $\mathbb{R}^n$. And the free $G$-action on $M$ is also given by polynomial equations of the $n$-variables of coordinates of $\mathbb{R}^n$. Could we use persistent homology and computer programming codes to compute the homology of $M/G$?

How to do it? Are there any programmes or codes that can be applied?