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A vector bundle is a $GL(\mathbb{R}^m)$-bundle with fiber $\mathbb{R}^m$. A principal $G$-bundle is a $G$-bundle with fiber $G$ (where the "$G$" in "$G$-bundle" embeds into $\mathrm{Aut}\, [G]$ by left-multiplication). In the former case, the appropriate notion of a map (at least for my purposes) is a bundle map that is linear on each fiber; in the latter case, the appropriate notion is a $G$-equivariant bundle map.

The question is: does there exist a general definition for a map between $G$-bundles that reduces to these two special cases when you take $G=GL(\mathbb{R}^m)$, $F=\mathbb{R}^m$, and $G=G$, $F=G$ respectively?

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These are two different things and one of them only makes sense when the total space has a fiber-preserving $G$-action. A bundle automorphism of a $G$-bundle $E \longrightarrow M$ is a diffeomorphism $\varphi: E \longrightarrow E$ that preserves fibers. A gauge transformation of a principal $G$-bundle $P \longrightarrow M$ is a bundle automorphism $\varphi: P \longrightarrow P$ that is equivariant with respect to the $G$-action on $P$: $$\varphi(p.g) = \varphi(p).g \text{ for all } g \in G.$$ Note that the definition of a gauge transformation doesn't make sense for general $G$-bundles. In general, a $G$-bundle only has a $G$-action defined on the model fiber $F$ (which we can only identify with a fiber using a local trivialization!), but not on the total space. A principal $G$-bundle is a special type of $G$-bundle with a nice $G$-action on the total space. Also note that if $F = G$, the bundle might still not have any fiber-preserving $G$-action on the total space. Hence the case $F = G$ is still not "special" enough.

If you restrict your attention to $G$-bundles with a fiber-preserving $G$-action on the total space, then the definition of gauge transformation will still make sense. In category theoretic terms, you should consider the category of principal $G$-bundles over $M$ as a non-full subcategory of the category of $G$-bundles with fiber $G$ over $M$.

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