We present the adiabatic quantum dynamics of the proton-transfer reaction H(S-2) + HeH+(X-1 Sigma(+)) -> H-2(+)(X-2 Sigma(+)(g)) + He(S-1) on the HeH2+ (X) over tilde (2)Sigma(+) RMRCI6 (M = 6) PES of C. N. Ramachandran et al. (Chem. Phys. Lett. 2009, 469, 26). We consider the HeH+ molecule in the ground vibrational rotational state and obtain initial-state-resolved reaction probabilities and the ground-state cross section sigma(0) and rate constant k(0) by propagating time-dependent, coupled-channel, real wavepackets (RWPs) and performing a flux analysis. Three different wavepackets are propagated to describe the wide range of energies explored, from cold (0.0001 meV) to hypertherrnal (1000 meV) collision energies, and in a temperature range from 0.01 to 2000 K We compare our time-dependent results with the time-independent ones by D. De Fazio and S. Bovino et al., where De Fazio carried out benchmark coupled-channel calculations whereas Bovino et al. employed the negative imaginary potential and the centrifugal-sudden approximations. The RWP cross section is in good agreement with that by De Fazio, except at the lowest collision energies below similar to 0.01 meV, where the former is larger than the latter. However, neither the RWP and De Fazio results possess the huge resonance in probability and cross section at 0.01 meV, found by Bovino et al., who also obtained a too low sigma(0) at high energies. Therefore, the RWP and De Fazio rate constants compare quite well, whereas that by Bovino et al. is in general lower.