In a frame moving relative to the earth, one has to include the kinetic energy of the earth as well as earth’s acceleration to be able to apply conservation of energy to the problem. In a reference frame falling to the earth with velocity vo, the stone is initially going up with velocity vo and so is the earth. The final velocity of the stone is 0 = vo- gt and that of the earth is vo+m/Mgt (M is the mass of the earth), from Newton’s third law, where t = time of fall. From conservation of energy
The point is this in earth’s rest frame the effect of earth’s acceleration is of order m/M and can be neglected but in a frame moving with respect to the earth the effect of earth’s acceleration must be kept because it is of order one (i.e. large).