Let $\pi\colon\tilde{X}\to X$ be a pfold (regular) cyclic covering(p:prime) and $\mathcal{A} = \mathrm{Im}(\pi_* )$, where $\pi_* \colon H_1(\tilde{X};\mathbb{Z}_p) \to H_1(X;\mathbb{Z}_p)$ is induced by $\pi\colon\tilde{X}\to X$. Suppose $f \colon X\to X$ is a homoemorphism satisfying $f_*(\mathcal{A})=\mathcal{A}$, where $f_* \colon H_1(X;\mathbb{Z}_p)\to H_1(X;\mathbb{Z}_p)$ induced by $f\colon X\to X$. Then, does there exists a map $g\colon \tilde{X}\to \tilde{X}$ satisfying $\pi\circ g =f\circ \pi$? I know the lifting criteria of covering space in terms of fundamental group. I hope that somebody tell me why above condition implies the lifting criteria in terms of fundamental group.
