A bstract Results are presented from a search for the Higgs boson decay H → Zγ, where Z → ℓ + ℓ − with ℓ = e or μ. The search is performed using a sample of proton-proton (pp) collision data at a center-of-mass energy of 13 TeV, recorded by the CMS experiment at the LHC, corresponding to an integrated luminosity of 138 fb − 1 . Events are assigned to mutually exclusive categories, which exploit differences in both event topology and kinematics of distinct Higgs production mechanisms to enhance signal sensitivity. The signal strength μ , defined as the product of the cross section and the branching fraction $$ \left[\sigma \left(\textrm{pp}\to \textrm{H}\right)\mathcal{B}\left(\textrm{H}\to \textrm{Z}\upgamma \right)\right] $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mfenced><mml:mrow><mml:mi>σ</mml:mi><mml:mfenced><mml:mrow><mml:mi>pp</mml:mi><mml:mo>→</mml:mo><mml:mi>H</mml:mi></mml:mrow></mml:mfenced><mml:mi>B</mml:mi><mml:mfenced><mml:mrow><mml:mi>H</mml:mi><mml:mo>→</mml:mo><mml:mi>Zγ</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced></mml:math> relative to the standard model prediction, is extracted from a simultaneous fit to the ℓ + ℓ − γ invariant mass distributions in all categories and is measured to be μ = 2 . 4 ± 0 . 9 for a Higgs boson mass of 125.38 GeV. The statistical significance of the observed excess of events is 2.7 standard deviations. This measurement corresponds to $$ \left[\sigma \left(\textrm{pp}\to \textrm{H}\right)\mathcal{B}\left(\textrm{H}\to \textrm{Z}\upgamma \right)\right]=0.21\pm 0.08 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mfenced><mml:mrow><mml:mi>σ</mml:mi><mml:mfenced><mml:mrow><mml:mi>pp</mml:mi><mml:mo>→</mml:mo><mml:mi>H</mml:mi></mml:mrow></mml:mfenced><mml:mi>B</mml:mi><mml:mfenced><mml:mrow><mml:mi>H</mml:mi><mml:mo>→</mml:mo><mml:mi>Zγ</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:mn>0.21</mml:mn><mml:mo>±</mml:mo><mml:mn>0.08</mml:mn></mml:math> pb. The observed (expected) upper limit at 95% confidence level on μ is 4.1 (1.8), where the expected limit is calculated under the background-only hypothesis. The ratio of branching fractions $$ \mathcal{B}\left(\textrm{H}\to \textrm{Z}\upgamma \right)/\mathcal{B}\left(\textrm{H}\to \upgamma \upgamma \right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>B</mml:mi><mml:mfenced><mml:mrow><mml:mi>H</mml:mi><mml:mo>→</mml:mo><mml:mi>Zγ</mml:mi></mml:mrow></mml:mfenced><mml:mo>/</mml:mo><mml:mi>B</mml:mi><mml:mfenced><mml:mrow><mml:mi>H</mml:mi><mml:mo>→</mml:mo><mml:mi>γγ</mml:mi></mml:mrow></mml:mfenced></mml:math> is measured to be $$ {1.5}_{-0.6}^{+0.7} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mn>1.5</mml:mn><mml:mrow><mml:mo>−</mml:mo><mml:mn>0.6</mml:mn></mml:mrow><mml:mrow><mml:mo>+</mml:mo><mml:mn>0.7</mml:mn></mml:mrow></mml:msubsup></mml:math> , which agrees with the standard model prediction of 0 . 69 ± 0 . 04 at the 1.5 standard deviation level.