Motivated by recent attempts to quantum simulate lattice models with continuous Abelian symmetries using discrete approximations, we define an extended-O(2) model by adding a γ cos (q φ ) term to the ordinary O(2) model with angular values restricted to a 2 π interval. In the γ →∞ limit, the model becomes an extended q -state clock model that reduces to the ordinary q -state clock model when q is an integer and otherwise is a continuation of the clock model for noninteger q . By shifting the 2 π integration interval, the number of angles selected can change discontinuously and two cases need to be considered. What we call case 1 has one more angle than what we call case 2. We investigate this class of clock models in two space-time dimensions using Monte Carlo and tensor renormalization group methods. Both the specific heat and the magnetic susceptibility show a double-peak structure for fractional q . In case 1, the small-β peak is associated with a crossover, and the large-β peak is associated with an Ising critical point, while both peaks are crossovers in case 2. When q is close to an integer by an amount Δ q and the system is close to the small-β Berezinskii-Kosterlitz-Thouless transition, the system has a magnetic susceptibility that scales as ∼1 /(Δ q )1-1 /δ' with δ' estimates consistent with the magnetic critical exponent δ =15 . The crossover peak and the Ising critical point move to Berezinskii-Kosterlitz-Thouless transition points with the same power-law scaling. A phase diagram for this model in the (β ,q ) plane is sketched. These results are possibly relevant for configurable Rydberg-atom arrays where the interpolations among phases with discrete symmetries can be achieved by varying continuously the distances among atoms and the detuning frequency.