In the Atiyah-Singer paper mentioned above, they introduced on p.557 a concept called $H$-structure which is used to describe the Chern character of special elements of $K(TX)$. It is roughly the following: Take a principal $H$-bundle $P$, so that $P \times_H V \cong TX$ for some $V$ and use this to construct a map from $K_G(V)$ to $K(TX)$.

On p. 559, Remark 2, they describe how this concept can be generalised to $H$-structures with an action of a group $G$. For the special case that $G$ acts trivially on the manifold $X$, they state that

"the action of $G$ on the principal $H$-bundle $P$ is given by some homomorphism $\rho:G \to$ Centre($H$)"

I actually do not believe this is true (or I misunderstood what kind of actions are considered here). In my opinion we only have fibre-preserving left action of $G$ on $P$, which can be given by any map $G \to H$ (we assume that the base space $X$ is connected).

Can somebody familiar with the paper give me argument for (or against) it?

PS: Any source where the case of Remark 2 is calculated in special cases would also be helpful.