@MikeMiller My initial idea for
this problem was to lift loops in $X$ based at $x_0 \in A$ to $\widetilde{X}$, and by simple connectedness homotope the lifted path to some path in $p^{-1}(A)$, and finally pushing down that homotoped path to a loop in $A$ based at $x_0$. This, I think, constructs a well-defined map $\pi_1(X) \to \pi_1(A)$ which is probably the right inverse of the map $i_\#$.