I am able to do part a) and I believe it should be used in solving part b). I think that for part b) we should that $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$, $G: \mathbb{R}^n \rightarrow \mathbb{R}^n$ so $F \circ G: \mathbb{R}^n \rightarrow \mathbb{R}^n$. However I cannot see how to go much further than this.
Note, or recall, that $X(x) = \frac{d}{dt}|_{t=0}\Phi_t(x).$ That's a basic fact about the correspondence between vector fields and flows. Now, put $\Gamma_t := G \circ \Phi_t \circ G^{-1}.$ Then, according to part (a) of the exercise, $\Gamma_t$ is a flow. Denote the vector field corresponding to $\Gamma_t$ by $Y.$ From definitions and our constructions, we get $$ Y(x) = \frac{d}{dt}|_{t=0}\Gamma_t(x) = \frac{d}{dt}|_{t=0} (G \circ \Phi_t \circ G^{-1})(x) = (G_*X)(x). $$ With that, we conclude $$ \begin{align} ((F\circ G)_*X)(x) &= \frac{d}{dt}|_{t=0}(F\circ G \circ \Phi_t\circ G^{-1}\circ F^{-1})(x) \\ & = \frac{d}{dt}|_{t=0}(F\circ \Gamma_t\circ F^{-1})(x) \\ & = (F_*Y)(x) \\ & = (F_*(G_*X))(x), \end{align} $$ as desired.