Let $M$ denote a closed manifold. Let $\Gamma(TM\setminus 0) $ denote the space of non-vanishing sections of $TM$. Note that the diffeomorphism group $\text{Diff} (M)$ acts on $\Gamma(TM\setminus 0) $ via $f. \phi(x) =Df_{f^{-1}(x)}(\phi(f^{-1}(x))$. I'm interested in the induced action on $\pi_0( \Gamma(TM\setminus 0))$, which evidently factors through $\pi_0(\text{Diff}(M))$.

The most interesting question for me at the moment is whether this induced action is faithful. Especially whether it is faithful if $M$ is an orientable $3$-manifold. But I'm also interested in more facts about this action and what kind of methods are helpful in understanding this action.