We study various types of mean equicontinuity and mean sensitivity for actions of countable discrete amenable groups. First, we show that \(\mathcal {F}\) -mean equicontinuity and \(\mathcal {F}\) -equicontinuity in the mean are equivalent when \(\mathcal {F}\) is a two-sided Følner sequence for a countable discrete amenable group. We then prove that, for \(\mathcal {F}\) -mean equicontinuity and \(\mathcal {F}\) -mean sensitivity, the Hausdorff and uniform versions are equivalent on compact Hausdorff spaces and to the classical definitions on compact metrizable spaces. Finally, for point-transitive dynamical systems, we establish a dichotomy theorem between uniform \(\mathcal {F}\) -mean equicontinuity and uniform \(\mathcal {F}\) -mean sensitivity.
