I'm not exactly a differential geometer, so I hope this isn't too elementary a question.

From a naive point of view, it seems as if there are two natural group actions on the space of connections on the tangent bundle of a manifold $X$: the group of diffeomorphisms, which acts by pull-backs, and the group of gauge transformations, which acts by, well, gauge transformations.

Now, there isn't any obvious relationship between the two groups and, as far as I can tell, the gauge group is a much more natural and much more convenient object to study.

However, on the other hand, it is certainly possible that for given diffeomorphism $\phi: X \to X$ and a connection $A$, we have a gauge equivalence between $\phi^{*}(A)$ and $A$. I'm interested in understand when this occurs in general. In particular, for which diffeomorphisms is it true that $\phi^*(A)$ is gauge equivalent to $A$, for *every* connection $A$?

I tried attacking this, at least for diffeomorphisms isotopic to the identity, by viewing the isotopy as coming from a flow on $X$ and writing down a differential equation satisfied by a one-parameter family of gauge transformations inducing the same map on some fixed connection, but it didn't seem to lead anywhere.

Thanks.