Is the tangent bundle of a covered manifold a quotient manifold?

Question

Given a covering manifold $\rho :\widetilde M \to M$ we know that $M$ can be thought of as the quotient space of $\widetilde M$ like so $M = \widetilde M /\ G$ where $G$ is the monodromy group (or fundamental group of $\widetilde M$). Is there a way to relate $T\widetilde M$ and $TM$ using a quotient construction?

Answer 1

Assuming $M$ is a smooth manifold and $G$ acts on $\widetilde{M}$ by diffeomorphisms, the $G$-action lifts naturally to $T\widetilde{M}$ (with elements of $G$ acting by the push-forward on tangent spaces), and $TM = T\widetilde{M}/G$.