  We perform a detailed study of a specific Two Higgs Doublet Model (2HDM) with
a U(1) gauge symmetry, instead of a typical Z2 discrete symmetry, containing a
very light gauge boson Z' (GeV scale or below). The Standard Model (SM)
fermions do not carry U(1) charges, but induced couplings to the Z' (called the
dark Z) are generated through mixing with the SM neutral gauge bosons. Such a
light Z' could explain some astrophysical anomalies as well as the muon g-2
deviation, and has been the subject of great experimental interest. We consider
the scenario in which the 125 GeV SM-like Higgs (H) is the heavier scalar
state, and focus on the lighter neutral state (h) as well as charged Higgs. We
analyze the constraints on the model from various experiments and predict novel
channels to search for these Higgs scalars at the LHC. In particular,
experiments looking for lepton-jets are among potentially important searches.
