  Multi-Higgs models equipped with global symmetry groups, either exact or
softly broken, offer a rich framework for constructions beyond the Standard
Model and lead to remarkable phenomenological consequences. Knowing all the
symmetry options within each class of models can guide its phenomenological
exploration, as confirmed by the vast literature on the two- and
three-Higgs-doublet models. Here, we begin a systematic study of finite
non-abelian symmetry groups which can be imposed on the scalar sector of the
four-Higgs-doublet model (4HDM) without leading to accidental symmetries. In
this work, we derive the full list of such non-abelian groups available in the
4HDM that can be constructed as extensions of cyclic groups by their
automorphism groups. This list is remarkably restricted but it contains cases
which have not been previously studied. Since the methods we develop may prove
useful for other classes of models, we present them in a pedagogical manner.
