Remark:

If $\phi : \widetilde X \longrightarrow \widetilde X$ is a continuous map such that $p \circ \phi = p$, then prove that $\phi$ is a homeomorphism in the following cases:

(i) $\pi_1(\widetilde X)$ is a finite group (ii) $p_{*}\pi_1(\widetilde X, \tilde x_0)$ has finite index in $\pi_1(X, x_0)$ (iii) $\widetilde X$ is a regular cover of $X$. Is this true in general? The point is that if $H$ is a subgroup of $G$ and $gHg^{-1} \subset H$ then it follows $gHg^{-1} = H$ in case $H$ is finite or has finite index or is normal.