Quotient of tangent bundle

Question

Let $M\to B$ be a principal $G$-bundle. Then I have a group action of $G$ on $M$. I think, that this induces a group action of $G$ on the tangent bundle $TM$ of $M$ by looking at the differentials of the maps $M\ni x\to x\cdot g\in M$ for each $g\in G$ (Am I right?). Can we view the space $TM/G$ as a vector bundle over $B$? If yes, how exactly?