The circle $S^1$ has fundamental group $\pi_1(S^1)=\mathbb{Z}$, universal cover $\mathbb{R}$, and satisfies $S^1=\mathbb{R}/\pi_1(S^1)$.
Similarly, $\mathbb{R}P^2$ has fundamental group $\pi_1(\mathbb{R}P^2)=\mathbb{Z}/2\mathbb{Z}$, universal cover $S^2$, and satisfies $\mathbb{R}P^2=S^2/(\mathbb{Z}/2\mathbb{Z})$.
Question: Let $M$ be a topological space with universal cover $\overline{M}$. Under what conditions is $$M=\overline{M}/\pi_1(M)?$$