The question is rather open-ended but I hope it is concrete enough.
If $M$ be a closed parallelizable smooth manifold with a smooth properly discontinuous co-compact action of a Lie group $G,$ what are the obstructions to $M/G$ being parallelizable?
This question was motivated by looking at the Hopf fibrations $S^1\to S^3\to S^2$ and $S^3\to S^7\to S^4$, where the total space is parallelizable but the base are not. It would also be nice to see what are these obstructions in the cases of the fibrations $O(n)\to O(n+1)\to S^n$ and how do they change depending on $n$.
Any insights are welcome. Thank you very much.