Morphism of associated bundles induces morphism of principial bundles

Question

Let $P$ be a principial $G$ bundle and $P'$ be a principial $G'$ bundle, let also $\rho:G \to G'$ be a group homomorphism and let us assume that $G' \subset GL(n)$. Let us consider vector bundles (the so called associated bundles) $P \times_{\rho} \mathbb{R}^n$ and $P' \times_{id} G'$ and lets make an assumption that these two vector bundles are isomorphic. Is it true that in this case we get a map of principial bundles $\eta:P \to P'$ (meaning that is acts fiberwise and is equivariant $\eta(pg)=\eta(p)\rho(g)$? If it is not true in general I would be happy with very special situation of $G=spin(n), G'=SO(n), P'=SO(M)$ (the frame bundle of the oriented riemanian manifold) and $\rho$ being the $2:1$ covering map.