Hopf actions and Nakayama automorphisms

Let H be a Hopf algebra with antipode S, and let A be an N-Koszul Artin–Schelter regular algebra. We study connections between the Nakayama automorphism of A   and S2S2 of H when H coacts on A inner-faithfully. Several applications pertaining to Hopf actions on Artin–Schelter regular algebras are given.