If $p: \widetilde{M} \to M$ is a covering map and $X$ a vector field on $M$, then exists a vector field satisfying $dp \widetilde{X} = X \circ h $

Question

I'm reading the book "Dynamical Systems on Surfaces - C.Godbillon", and on page 2, he claims the following result:

Let $\widetilde{M}$ and $M$ be smooth manifolds without boundary, $p: \widetilde{M}\to M$ a smooth covering map and $X$ a smooth vector field on $M$. Then there exists a uniquely defined smooth vector field $\widetilde{X}$ on $\widetilde{M}$ such that $$\text{d}p(x) \widetilde{X}(x) = X \circ p(x) $$

I would like to demonstrate this result (I only need the existence part), but I'm not getting much progress.

Can anyone help me (just a reference is enough)?