  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 Kwa\'{s}niewski 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.
