Is there any natural group structure on the set $I_p = \{x \in \mathbb{Z}[1/p] \mid x < p/2\}$?

This set doesn't have a subgroup structure as a subset of the reals, but if p is an odd prime, you can certainly lift the additive group law on Q_{p}/pZ_{p} to this set. An alternative way to look at it that works for arbitrary odd numbers p is to lift the ppower roots of unity in the complex numbers by the map x > e^{2 pi ix/p}. If p is even, you probably have to make more choices. 

