If a group $G$ acts on a Cantor set $(X,\mu)$ by odometers, my question is: what is the explicit automorphism $\alpha_{g}$ for the extended Koopman action on $L^{\infty}(X,\mu)$, for $g$ $\in$ $G$?  I am not getting by lots of trying; in many books it's written that it is adding by $(\cdots,0,0,1)$  on the sequence of 0,1's. Kindly help me figuring it out.