  In this work, we discuss a distinctive $pp\to {\rm Higgs}\to Z'Z'\to 4l$
($l=e,\mu$) signal at the Large Hadron Collider (LHC), where the `Higgs' label
refers to the SM-like Higgs state discovered in 2012 or a lighter one in the
framework of a theoretical model embedding a spontaneously broken $U(1)'$
symmetry in addition to the Standard Model (SM) gauge group. The additional
$U(1)'$ symmetry generates a very light $Z'$ state, with both vector and axial
(non-universal) couplings to fermions, which are able to explain the so-called
Atomki anomaly, compliant with current measurements of the Anomalous Magnetic
Moments (AMMs) of electron and muon as well as beam dump experiments. We show
that the cross section for this process should be sufficiently large to afford
one with significant sensitivity during Run 3 of the LHC.
