Vector Bundles, Principal Bundles and Bundle of Frames

Question

I have the following question. I have seen many times in the literature the following construction and I don't understand it too well, so I hope some of you can actually help me. For the sake of clarity, I will assume that everything works well in order to exist the quotient by groups action. Let us suppose that we have a principal $G$-bundle $P\rightarrow M$, and a representation of $G$ in a complex finite-dimensional vector space $V$, $\rho:G\rightarrow Aut(V)$. We can form the associated vector bundle $E:=(P\times V)/G$ through the action $g\cdot (p,v):=(p\cdot g, \rho(g)^{-1}(v))$, and we can form a $Aut(V)$ vector bundle, given by the extension of the structure group of $P$, this is $(P\times GL(V))/G$. In many texts, the authors use indistinctly $(P\times Gl(V))/G$ and the frame bundle of $E$, this is the $Aut(V)$-bundle $Isom_{M}(E, V\times M)$ where $V\times M$ is the trivial vector bundle of typical fiber $V$ over $M$. So my question is: what is the isomorphisms between the manifolds of isomorphism and $(P\times Gl(V))/G$.