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.