This should surely be well-known by I have not been able to find a good reference to the following question: Given a smooth fiber bundle $\pi\colon P \longrightarrow M$ over a smooth manifold $M$ with typical fiber $F$, one has the group of fiber-preserving automorphisms of $P$: a diffeomorphism $\Phi\colon P \longrightarrow P$ is called fiber-preserving if $\pi \circ \Phi = \phi \circ \pi$ for some smooth map $\phi\colon M \longrightarrow M$, which then turns out to be a diffeomorphism of $M$. If $\phi = id$ then one calls $\Phi$ a gauge transformation. Clearly they form a normal subgroup $\mathrm{Gau}(P) \subseteq \mathrm{Aut}(P)$, being the kernel of the group morphism $\Phi \mapsto \phi$. Hence we get a subgroup of the diffeomorphism group as the image of this quotient $\mathrm{Aut}(P) / \mathrm{Gau}(P) \subseteq \mathrm{Diffeo}(M)$.

It is now well-known and not too hard to show that all the small diffeomorphisms of $M$ are contained in this image: this can be done by using a (complete) connection and it's parallel transport.

My question is about the large diffeomorphisms: are they also in the image, i.e. is the whole diffeomorphism group isomorphic to this quotient $\mathrm{Aut}(P) / \mathrm{Gau}(P)$? What conditions of $P$ would guarantee this (beside being trivial...)?

This should surely be well-known by I have not been able to find a good reference to the following question: Given a smooth fiber bundle $\pi\colon P \longrightarrow M$ over a smooth manifold $M$ with typical fiber $F$, one has the group of fiber-preserving automorphisms of $P$: a diffeomorphism $\Phi\colon P \longrightarrow P$ is called fiber-preserving if $\pi \circ \Phi = \phi \circ \pi$ for some smooth map $\phi\colon M \longrightarrow M$, which then turns out to be a diffeomorphism of $M$. If $\phi = \mathrm{id}_M$ then one calls $\Phi$ a gauge transformation. Clearly they form a normal subgroup $\mathrm{Gau}(P) \subseteq \mathrm{Aut}(P)$, being the kernel of the group morphism $\Phi \mapsto \phi$. Hence we get a subgroup of the diffeomorphism group as the image of this quotient $\mathrm{Aut}(P) / \mathrm{Gau}(P) \subseteq \mathrm{Diffeo}(M)$. Of course, the case of principal fiber bundles is of particular interest here.

It is now well-known and not too hard to show that all the small diffeomorphisms of $M$ are contained in this image: this can be done by using a (complete) connection and it's parallel transport.

My question is about the large diffeomorphisms: are they also in the image, i.e. is the whole diffeomorphism group isomorphic to this quotient $\mathrm{Aut}(P) / \mathrm{Gau}(P)$? What conditions of $P$ would guarantee this (beside being trivial...)?