Principal bundle automorphism

Let $\pi:P\to M$ be a principal G-bundle .

A principal bundle automorphism is a principal bundle isomorphism $\Phi:P\to P$ (so $\Phi$ is an equivariant diffeomorphism) covering a base diffeomorphism $f:M\to M$, meaning

The automorphisms of $P$ form a group under composition, often denoted $\mathrm{Aut}(P)$. The subgroup consisting of automorphisms with $f=\mathrm{id}_M$ is the group of gauge transformations .

Examples

1. Automorphisms of a trivial bundle. If $P=M\times G$, any pair consisting of a diffeomorphism $f:M\to M$ and a smooth map $a:M\to G$ defines $\Phi(x,h)=(f(x),\,a(x)\,h),$ which is equivariant and hence an automorphism.
2. Lift of a base diffeomorphism to frames. A diffeomorphism $f:M\to M$ acts on the frame bundle by sending a frame at $x$ to its pushforward frame at $f(x)$; this is a principal bundle automorphism.
3. Central right multiplication. If $z$ lies in the center of $G$, then $\Phi(p)=p\cdot z$ is equivariant and defines an automorphism covering $\mathrm{id}_M$.