  We characterize which mappings from a compact subset of $\mathbb{R}$ into the
Heisenberg group can be extended to a $C^{m,\omega}$ horizontal curve for a
given modulus of continuity $\omega$. We motivate our characterization by
showing that the $C^{m,\omega}$ extension property fails if we instead use a
more direct analogue of the conditions from the $C^{m}$ case.
