Many years ago Khalatnikov described unusual properties of the Doppler shift for the second sound in He II, first of all the ``back-entrainment'' effect: at some temperatures (at the beginning of the roton region) the center of the spreading sound moves in the direction opposite to the normal-component velocity ${\mathit{v}}_{\mathit{n}}$ [\ensuremath{\Delta}${\mathit{u}}_{2}$=\ensuremath{\gamma}(T)${\mathit{v}}_{\mathit{n}}$, \ensuremath{\gamma}(T\ensuremath{\sim}0.6 K)0]. However, the existing theory describes Doppler shift of the first and fourth sounds as a trivial, ``kinematic'' effect: the center of the spreading sound moves with the velocity of the liquid as a whole [\ensuremath{\Delta}${\mathit{u}}_{1,4}$\ensuremath{\approxeq}j/p=(1-${\mathrm{\ensuremath{\rho}}}_{\mathit{n}}$/\ensuremath{\rho})${\mathit{v}}_{\mathit{s}}$]. We show that the real situation is quite different. We find (1) the coefficient K in the Doppler-shift expression, \ensuremath{\Delta}${\mathit{u}}_{1,4}$=(1-${\mathit{K}}_{1,4}$${\mathrm{\ensuremath{\rho}}}_{\mathit{n}}$/\ensuremath{\rho})${\mathit{v}}_{\mathit{s}}$, substantially differs from the kinematic value K=1:\ensuremath{\Vert}K${\mathrm{\ensuremath{\Vert}}}_{\mathrm{max}}$ reaches some tens. (2) ${\mathit{K}}_{1}$(T) and ${\mathit{K}}_{4}$(T) have different (qualitatively opposite) nontrivial temperature dependences, in particular a high peak (modulo) at the beginning of the roton region. (3) ${\mathit{K}}_{1,4}$(T) can be negative: ${\mathit{K}}_{1}$0 in the region of the peak, ${\mathit{K}}_{4}$0 in the phonon region. This implies an ``outstripping'' effect: the center of the spreading sound moves faster than the flowing superfluid part of the liquid itself.