  We classify the finite groups whose non-linear irreducible characters that
are not conjugate under the natural Galois action have distinct degrees,
therefore extending the results in Berkovich et al. [Proc. Amer. Math. Soc.
{\bf 115} (1992), 955-959] and Dolfi et al. [Israel J. Math. {\bf 198} (2013),
283-331].
