In this paper, we are interested in the construction of quasiconformal mappings between domains of the Heisenberg group
H
\mathbf {H}
that minimize a mean distortion functional. We propose to construct such mappings by considering a corresponding problem between domains of Poincaré half-plane
H
\mathbb H
and then, lifting every of its solutions to
H
\mathbf H
. The first map we construct is a quasiconformal map between two cylinders. We explain the method used to find it and prove its uniqueness up to rotations. Then, we give geometric conditions which ensure that a minimizer (in
H
\mathbf {H}
) comes as a lift of a minimizer between domains of
H
\mathbb H
. Finally, as a non-trivial example of the generalization, we manage to reconstruct the map from [Ann. Acad. Sci. Fenn. Math. 38 (2013), pp. 149–180] between two spherical annuli and prove its uniqueness as a minimizer.