Modular curves of prime-power level with infinitely many quadratic points

We completely determine the 1085 open subgroups H of GL₂ (Z) of prime-power level that satisfy -I H and det (H) =Z^ for which the corresponding modular curve XH has infinitely many quadratic points. When g (XH) 2 this is equivalent to determining all the hyperelliptic modular curves of prime-power level and all the bielliptic modular curves of prime-power level that admit a degree two map to a positive rank elliptic curve. From the moduli perspective, this means that there are exactly 1085 subgroups H of GL₂ (Z) of prime-power level for which there are infinitely many elliptic curves E/K over quadratic extensions such that ρE (Gₖ) is conjugate to a subgroup of H.