Inclusive, differential and production mode cross sections of the Higgs boson production in proton-proton collisions are measured in the $H \to ZZ^{*} \to 4\ell$ decay channel. The proton$-$proton collision data were produced at the Large Hadron Collider at a centre-of-mass energy of 13 TeV and recorded by the ATLAS detector from 2015 to 2017, corresponding to an integrated luminosity of 79.8 fb$^{-1}$. The inclusive fiducial cross section for the process $H \to ZZ^{*} \to 4\ell$ is measured to be $\sigma_\mathrm{fid} = 4.04 \pm 0.47$ fb, while the Standard Model prediction is $\sigma_\mathrm{fid, SM} = 3.35 \pm 0.15$ fb. The cross-section times $H \to ZZ^{*}$ branching ratio for gluon fusion and vector-boson fusion production are measured to be $1.21 \pm 0.18$ pb and $0.24 \pm 0.09$ pb, respectively. All measurements are in agreement with the Standard Model prediction.

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