In §1.6 of Atiyah's *K-theory*, he defines the notion of a $G$-(vector)-bundle, which is a sort of "equivariant vector bundle" with respect to a finite group action. More specifically, let $G$ be a finite group and $X$ a compact Hausdorff space. A $G$-bundle (not in the principal sense) is a vector bundle $\pi:E\rightarrow X$, where both $E$ and $X$ are equipped with continuous $G$-actions, such that $\pi$ is equivariant and the fiber map $g:\pi^{-1}(x)\rightarrow \pi^{-1}(gx)$ is linear for every $g\in G$ and $x\in X$. A $G$-bundle is said to be trivializable if there is a $G$-representation $G\rightarrow \text{GL}(V)$ and a $G$-equivariant map $E\rightarrow X\times V$ (where $G$ acts component-wise on the product).


This is the "correct" notion, in the sense that a $G$-bundle over $X$ is trivializable if and only if it is isomorphic to the pullback of a $G$-bundle $V\rightarrow \text{pt}$ (this is the same thing as a $G$-representation on $V$) along the constant map $X\rightarrow \text{pt}.$ However, the local triviality assumed for $\pi:E\rightarrow X$ does not involve the group action and hence seems weaker. My question is: Is a $G$-bundle always locally trivial in this stronger sense? If not, do you know of a relatively simple counterexample?


My thoughts thus far, which may be wrong, are the following:


Such a local trivialization would need to be over a $G$-invariant neighborhood $U\subset X$. If we have a trivialization $\varphi:\pi^{-1}(U)\rightarrow U\times V$ in the usual (weaker) sense, then the $G$-action on $\pi^{-1}(U)$ defines an action on $U\times V$, which is essentially a $U$-parametrized family of $G$-representations on $V$, i.e. a map $\mu:U\times G\rightarrow \text{GL}(V)$ that is a homomorphism when we fix the first coordinate. To get a trivialization in the stronger sense, the goal is for $\mu$ to be constant in the first factor (we can shrink $U$, so long as it remains $G$-invariant). Then we can modify the trivialization to make this so if and only if there is a map $\lambda:U\rightarrow \text{GL}(V)$ such that
$\lambda(x)\mu(x,g)\lambda(x)^{-1}$ is independent of $x\in U$ (for each $g\in G$).


But my present knowledge of Lie groups is rather weak and I'm not sure how to proceed from here.



Locally trivializing a G vector bundle?
To get an idea of what is going on, suppose the base space $X$ was a single orbit of $G$, say $X = G/H$, with $H$ a subgroup. A $G$-equivariant vector bundle over $G/H$ will be isomorphic to one of the form $G \times\_H V \rightarrow G/H$, where $V$ is a representation of $H$. This will only be isomorphic to $G/H \times W \rightarrow G/H$, with $W$ a representation of $G$, if the $H$-representation $V$ was the restriction of some $G$-representation, which is not always the case.


