I came across the following example of a coordinate fibre in the book "An introduction to differential manifolds":
The (open) Mobius strip with $B=S^1$, $F=\mathbb{R}$, $G=\mathbb{Z}/2$. Here we identify $S^1$ with $\{z\in\mathbb{C}| |z|=1\}$, and take domain charts $U_1=S^1\setminus \{1\}$ and $U_2=S^1\setminus \{-1\}$. The intersection $U_1\cap U_2$ has two connected component, and the coordinate transformation $g_{21}$ is given by $g_{21}(x)=\pm1$ depending on the component in which $x$ lies.
My first question is: what are the homeomorphisms of $F$ that correspond to the elements of $G$? A natural choice seems to be $$G\ni 1\simeq f\mapsto f$$ $$G\ni -1\simeq f\mapsto -f$$ However, this would mean that on one connected component of $U_1\cap U_2$ we have $$\phi_{2,x}^{-1}\circ\phi_{1,x}=f\mapsto f$$ while on the other we have $$\phi_{2,x}^{-1}\circ\phi_{1,x}=f\mapsto -f$$ However, I'm having trouble coming up with concrete $\phi_1$ and $\phi_2$ that satisfy this.
So my question is: am I even interpreting the concept of a coordinate bundle correctly? And if I am, can someone please fill in the details by giving the $\phi_1$ and $\phi_2$ explicitly? Thanks.