Let $X$ be a compact, connected, locally path-connected, and semilocally simply connected topological group with $\pi_1(X) \cong \mathbb{Z}$.

Let $u\colon \widetilde{X}\to X$ be its topological universal cover, which also has the structure of a topological group.

Let $Y$ be a locally path-connected topological space.

Suppose there exists a surjective, local homeomorphism $f\colon \widetilde{X}\to Y$ and a (surjective) morphism $a\colon Y \to X$ with connected fibers such that $u = a\circ f$.

Thus, one could say that the universal cover $u$ ''factors'' through $f$ and $a$.

Question. With these above conditions, can one deduce that $Y$ is a topological group?

Thanks in advance for the help!

Let $X$ be a compact, connected, locally path-connected, and semilocally simply connected topological group with $\pi_1(X) \cong \mathbb{Z}$.

Let $u\colon \widetilde{X}\to X$ be its topological universal cover, which also has the structure of a topological group.

Let $Y$ be a locally path-connected topological space.

Suppose there exists a surjective, local homeomorphism $f\colon \widetilde{X}\to Y$ and a (surjective) morphism $a\colon Y \to X$ of topological spaces with connected fibers such that $u = a\circ f$.

Thus, one could say that the universal cover $u$ ''factors'' through $f$ and $a$.

Question. With these above conditions, can one deduce that $Y$ is a topological group?

Thanks in advance for the help!