I'm looking for a definition of the notion of isomorphism between vector bundles $E$ and $F$, over the same base $X$, whose structure groups have been reduced from $GL(n)$ to some subgroup $G\subseteq GL(n)$. For concreteness, let us work in the smooth category with $G$ a Lie group. This idea of classifying vector bundles with reduced structure group up to isomorphism seems to be an often-mentioned concept, and is closely related to the homotopy classes of maps $X\rightarrow BG$, and to the first Cech cohomology $H^1(X,G)$, but I have so far not seen it defined using just concepts from fibre bundles.
My attempt at the definition is to say that $E$ and $F$ are isomorphic if their underlying principal $G$-bundles are isomorphic, but I'm not sure how to formulate a definition without principal bundles.
The definition in https://en.wikipedia.org/wiki/Fiber_bundle#Bundle_maps does not apply to this case because $G$ may not act on the total space of $E$ and $F$. (See the first answer in https://mathoverflow.net/questions/92813/vector-bundles-vs-principal-g-bundles).
Remark 3.43 in https://www.lib.utexas.edu/etd/d/2008/klonoffk16802/klonoffk16802.pdf also hints at the connection to principal bundles, but again a definition of what it means for two Hilbert bundles $H_1$ and $H_2$, each with a reduction of its structure group to a subgroup of $GL(H)$ (where $H$ is the fibre), to be isomorphic, is not made explicit.
Any thoughts would be appreciated.