Induced group action on tangent bundle commutes with structure group?

Question

I am trying to understand how the free and proper action of a discrete group $\Gamma$ on a manifold $X$ by automorphisms changes the structure group of the tangent bundle $\mathcal{T}_X$ of $X$. Let $\gamma\in\Gamma$ be an element and consider the automorphism induced by $\gamma$: $x\mapsto\gamma\cdot x$. This induces a map between tangent spaces $d\gamma:\mathcal{T}_{X,x}\to\mathcal{T}_{X,\gamma\cdot x}$. Consider a local trivialization of $\mathcal{T}_X$: $\pi^{-1}(U)\cong U\times F$ where $\pi$ and $F$ denote the projection map and typical fiber respectively. The transition functions of $\mathcal{T}_X$, which take values in the structure group $G$, give gluing maps $(x,f)\mapsto (x,t(x)\cdot f)$, where $x\in U$, $f\in F$, and $t:U\to G$. The action of $\Gamma$ also gives rise to $(x,f)\mapsto (\gamma\cdot x,d\gamma(f))$ for each $\gamma\in\Gamma$. Is it true that if the actions of $\Gamma$ and $G$ on the tangent bundle commute, i.e., if $d\gamma(t(x)\cdot f)=t(\gamma\cdot x)\cdot d\gamma(f)$, then the tangent bundle of the quotient $X/\Gamma$ also has structure group $G$? How do I see this? And if the actions don't commute what is the structure group of the tangent bundle of the quotient? Thanks for the help!