On Fock covariance for product systems and the reduced Hao-Ng isomorphism problem by discrete actions

We provide a characterisation of equivariant Fock covariant injective representations for product systems. We show that this characterisation coincides with Nica covariance for compactly aligned product systems over right LCM semigroups of Kwaśniewski and Larsen, and with the Toeplitz representations of a discrete monoid of Laca and Sehnem. By combining with the framework established by Katsoulis and Ramsey, we resolve the reduced Hao-Ng isomorphism problem for generalised gauge actions by discrete groups.