I am wondering if the following is true.
Let $G$ be a Lie group, $V$ a vector space, $\rho$ a representation of $G$ on V, and $\pi: E\rightarrow M$ a vector bundle with fibre $V$. Does there exist a unique (up to principal bundle isomorphism) principal $G$-bundle, such that $E\cong P\times_\rho V$, where $P\times_\rho V$ is the vector bundle associated to $P$ and the representation $\rho$ on $V$?
I know that every $\mathbb{K}$ vector bundle of rank $n$ is associated to a principal $\mathrm{GL}(n,\mathbb{K})$ bundle, but is this unique up to isomorphism? Is this also true if we have any subgroup of $\mathrm{GL}(n,\mathbb{K})$? Or even any Lie group with representation on the fibre vector space?