  The ATLAS and CMS collaborations at the LHC have recently announced evidence
for the rare Higgs boson decay into a $Z$ boson and a photon. We analyze the
interference between the process $gg\! \to \! H \! \to \! Z \gamma$ induced by
loops of heavy particles, which is by far the dominant contribution to the
signal, and the continuum $gg \to Z \gamma$ QCD background process mediated by
light quark loops. This interference modifies the event yield, the resonance
line-shape and the apparent mass of the Higgs boson. We calculate the radiative
corrections to this interference beyond the leading-order approximation in
perturbative QCD and find that, while differing numerically from the
corresponding effects on the more studied $gg \! \to \! \gamma \gamma$ signal,
they are generally rather small. As such, they do not impact significantly the
interpretation of the present measurements of the $H \to Z \gamma$ decay mode.
