In this answer a possible derivation of the group velocity is provided.
It is, anyway, based on the assumption that there will always be a point where all the cosines will sum with the same phase:
The peak will still be where the phases of the component waves are the same.
But if the waves travel at different velocities, the existence of such a peak is unlikely. So, what are the particular conditions (hypotheses) when the above statement is valid and the computation in the linked answer is acceptable?
What are the particular cases when all the waves within a certain $k$-range will always sum with the same phase in some points, during their propagation along $x$?