It is given in Progress in mathematics 62, Guillou and Marin book. In the proof of Lemma 4, They choose $\alpha$ and $r\in \mathbb{N}$ such that $h^r_*\colon H_1(X;Z_p)\to H_1(X;Z_p)$ satisfies $h^r_*(\alpha)=\alpha$ and conclude that $X\to X$ can be lifted to $\tilde{X}\to \tilde{X}$. Since $\tilde{X}$ is a convering space of $X$, I agree that it is enought to show that the primary obstruction $[\theta(h^r\circ p)]\in H^1(X;Z_p)$ vanishes. (Is this local coefficient system is trivial?) Why $h^r_*\colon H_1(X;Z_p)\to H_1(X;Z_p)$ satisfying $h^r_*(\alpha)=\alpha$ ensures that the existence of lifting $\tilde{h}\colon \tilde{X}\to \tilde{X}$ and why $\tilde{X}\to \tilde{X}/\tilde{h}$ is infinite covering?

Thank you