Are spaces $X, Y$ with $\pi_{1}(X) \cong \pi_{1}(Y)$, $\widetilde{X} \simeq \widetilde{Y}$ necessarily homotopy equivalent?

Question

Let $X,Y$ be path-connected, locally path-connected, semilocally simply-connected spaces with isomorphic fundamantal groups and homotopy equivalent universal covers. Are $X$ and $Y$ necessarily homotopy equivalent?

Answer 1

The lens spaces $L(p,q)$ all have $\mathbb{Z}/p\mathbb{Z}$ as fundamental groups and their universal cover is $S^3$. But $L(p,q_1)$ is homotopically equivalent to $L(p,q_2)$ if and only if $q_1q_2\equiv \pm n^2 \mod{p}$ for some $n$. Thus $L(5,2)$ and $L(5,4)$ provide a counterexample to your claim, since $2\cdot 4 \equiv 3 \mod 5$, but $\pm$squares can only be $1$ or $4$ $\mod 5$ (note that $-1=4$ and, of course, $-4=1$).