Let $\Gamma$ and $G$ be compact Lie groups and $\alpha:\Gamma\to Aut(G)$ group homomorphism with the condition that $(\gamma,g)\mapsto \alpha(\gamma)(g)$ is continuous.
$(\Gamma,\alpha,G)$-bundles are defined by tom Dieck in his book Transformation groups as follows (I.(8.7)):
A locally trivial $G$-principal bundle $\pi:P\to X$ together with left $\Gamma$-actions on $P$ and $X$ such that:
i) $p$ is $\Gamma$-equivariant
ii) For $\gamma\in\Gamma$, $g\in G$ and $p\in P$ the relation $\gamma\cdot(pg)=(\gamma p)\cdot\alpha(\gamma)(g)$ holds.
The total space $P$ carries a left action of the semi-direct product $\Gamma\times_\alpha G$.
At this stage, I wonder how an associated fibre bundle with fibre $F$ can be defined. In the non-equivariant case, $F$ s assumed to be a left $G$-space, and then $E=F\times_G P$, but I do not know if now $F$ should be considered to be (right, since the semi-direct product acts on the left on $P$) $(\Gamma\times_\alpha G)$-space.
I am particularly interested in the case $\Gamma=\mathbf Z/2\mathbf Z$ and $G=U(n)$, where $\alpha(\tau)(A)=A^\dagger$ ($\tau$ is the non-trivial element of $\mathbf Z/2\mathbf Z$), which reproduces the real vector bundles introduced by Atiyah in his paper K-theory and reality (Example 8.8 in tom Dieck's book).
I would like to get a better understanding of real vector bundles, and since I have not found any reference which provides a detailed description of the bundle charts or cocycles, I thought about trying approaching this way. Of course, references for real vector bundles (or the more general notion of $G$-equivaraint vector bundle described in I.9 of tom Dieck's book) are very welcome.