Let $(X,A)$ and $(Y,B)$ pairs satisfying the homotopy extension property (or a CW-pairs if necessary) with $A$ and $B$ contractible. Let $f:(X,A)\to (Y,B)$ a map of pairs and let $p_X:\widetilde{X}\to X$ and $p_Y:\widetilde{Y}\to Y$ covering maps.
By definition of covering space, every point $x\in X$ has a neighbourhood $U$ such that $p^{-1}(U)$ is a disjoint union of open sets each of them mapped homeomorphically onto $U$ by $p$, so in particular $p^{-1}(x)$ is a discrete set with one point on each component of $p^{-1}(U)$. By the theory of covering spaces we know that under certain conditions, a map $(X,x_0)\to (Y,y_0)$ may lift to $(\widetilde{X},\tilde{x}_0)\to (\widetilde{Y},\tilde{y}_0)$.
I am interested in knowing if such property are satisfied in my setting, namely, is $p^{-1}(A)$ a disjoint union of subspaces of $\widetilde{X}$ each of them mapped homeomorphically by $p$ onto $A$.
If so, I can choose $\widetilde{A}$ and $\widetilde{B}$ to be such subspaces of $\widetilde{X}$ and $\widetilde{Y}$ respectively. Then I wonder if the map $f:(X,A)\to (Y,B)$ can be (uniquely) lifted to $\tilde{f}:(\widetilde{X},\widetilde{A})\to (\widetilde{Y}, \widetilde{B})$ making the diagram of maps of pairs and covering maps commute.