Finite cover gives a lift of discrete Morse function

Question

Let's say I have a finite simplicial complex $X$ with a finite covering map $\pi: \widetilde{X} \rightarrow X$ and a discrete gradient vector field $V$ on $X$ (which for my purposes I prefer to its equivalent discrete Morse function). I need the lift $\widetilde{V}$ to be a discrete gradient vector field on $\widetilde{X}$. I believe it's true, and it's not hard to prove, but I would like to state a reference if such a thing exists.