## Possibilities for $\pi_1(\tilde{X})$ when $\pi_1(X)$ given

What are the possibilities for $$\pi_1(\tilde{X})$$ when $$\pi_1(X)$$ is given? Say $$\pi_1(X)\cong \mathbb{Z}_4$$. I can’t come up with a way to relate the fundamental groups of the two.