Inclusive, differential and production mode cross sections of the Higgs boson are measured in the $H \to ZZ^{*} \to 4\ell$ decay channel. Additionally, cross sections are measured for the main production modes in several exclusive regions of the Higgs boson production phase space. The dataset used corresponds to proton-proton collision data produced at the Large Hadron Collider at a centre-of-mass energy of 13 TeV and recorded by the ATLAS detector from 2015 to 2018, equivalent to an integrated luminosity of 139 fb$^{-1}$. The inclusive fiducial cross section for the process $H \to ZZ^{*} \to 4\ell$ is measured to be 3.35 $\pm$ 0.32 fb, while the Standard Model prediction is 3.41 $\pm$ 0.18 fb. The cross-section times $H \to ZZ^{*}$ branching ratio with $|y_H|<$~2.5 for gluon fusion and vector-boson fusion production are measured to be 1.14~ \pm ~0.13~ pb and 0.13 ~\pm ~0.04~ pb, respectively. All measurements are in agreement with the Standard Model prediction.

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