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Lifting a graph "folding" to a covering space

I have the following problem, which seems really simple but I can't seem to quite get it.

The problem is about connected edge labeled directed graphs. I like to think of these as an ordinary graph $K$ plus a graph homomorphism $\ell_K : K \rightarrow R_X$ from $K$ to a rose graph. Edges are then labeled and directed by which "petal" the map onto.

Then we say a folding $f : K \rightarrow K'$ is surjective graph homomorphism such that $\ell_K(\pi_1(K)) = \ell_{K'}(\pi_1(K'))$. That is to say it preserves the image of the fundamental group's projection.

We then say a normal cover is a covering map $p : \tilde A \rightarrow A$ such that the preimage of every point is a transitive set under deck transformations.


Now here is what I believe to be true and wish to show:
* I have normal covering map $p : \tilde A \rightarrow A$.
* I have a folding $f : A \rightarrow B$

There then exists a $\tilde B$, a normal covering map $p^\ast : \tilde B \rightarrow B$, and a folding map $f^\ast : \tilde A \rightarrow \tilde B$ such that $f \circ p = p^\ast \circ f^\ast$.

Summarized in a commutative diagram:

![Commutative diagram](https://math.codidact.com/uploads/eynnwtnnoqrw5ejrhnljw4cmk3ah)

This diagram indicates that we are constructing a bundle morphism. In some sense we are saying if I have a folding between the base spaces then there is a "bundle folding" between normal covers.

This seems in some sense "obvious", you simply run the same folding process in the covering space. But I can't quite make that formal. I can prove the finite case with some induction, but it seems far from proving the infinite case. I feel like I am missing some obvious perspective. A point in the right direction or a good reference to read would be very helpful.