Let $P$ be a principal $G$-bundle over $B$, and let $G$ act on some space $F$ (feel free to work in your favorite category of spaces, if this helps). Then $\text{Aut}{P}$ (aka the group of gauge transformations of $P$) acts on the space of sections of the associated bundle $P\times_G F$ as follows: if $\alpha\in\text{Aut}P$ then for each section $s\colon B\to P\times_G F$ we define $(\alpha\cdot s)(x)=[(\alpha u_x,f)]$ where $s(x)=[(u_x,f)]$ (since $\alpha$ is equivariant, this definition is independent of the representative $(u_x,f)$ for $s(x)$).
For brevity let us agree to denote by $a$ the action of $G$ on $F$, and by $A$ the action of $\text{Aut}{P}$ on $P\times_G F$.
My questions are the following: what properties of $a$ pass to $A$, in general (i.e. for any $P$)? In particular, what about transitivity?
It might be worthwhile to check a few special cases.
- If $a$ is the trivial action then $A$ is also trivial.
- If $a$ is the action of $G$ on itself by conjugation then $A$ is (up to some isomorphism) the action of $\text{Aut}P$ on itself by conjugation (where the multiplication is composition of automorphisms).
- If $a$ is the action of $G$ on itself by multiplication and $P$ is trivial then $A$ is (again up to some isomorphism) the action of $\text{Aut}P$ on itself by composition (here the case $P$ non-trivial implies $P\times_G G\simeq P$ has no section, making the question moot).
In all these cases it can be seen that transitivity, freeness, faithfulness pass from $a$ to $A$.
EDIT: I split the question in two, so the part dealing with the case $P$ trivial has now become this other question.