Let $M$ be a manifold and let $G$ be a finite group. Let $P \to M$ be a principal $G$-bundle. I have some basic questions regarding the automorphism group of $P$ as a principal $G$-bundle over $M.$. These are basically sanity checks. I do not need a proof, if what I am saying is true actually is true, just a confirmation. If the statements are false however, I would appreciate the correct statements.
- If $M$ is a point and $P$ is the trivial $G$-bundle, is it true that $Aut(P) \cong G?$.
- Let $M$ be the circle and let $P$ be a principal bundle corresponding to a morphism $f:\pi_1(M,m) \to G.$ Such a morphism gives an element $g \in G.$ Is it true that $Aut(P) \cong C_G(g)?$ Here $C_G(g)$ is the centralizer of $g$ in $G.$.