$G$-covariant $\phi$-coordinated quasi modules for quantum vertex algebras

This is a paper in a series to study quantum vertex algebras and their relations with various quantum algebras. In this paper, we introduce a notion of T-type quantum vertex algebra and a notion of $G$-covariant $\phi$-coordinated quasi module for a $T$-type quantum vertex algebra with an automorphism group $G$. We refine and extend several previous results and we obtain a commutator formula for $G$-covariant $\phi$-coordinated quasi modules. As an illustrating example, we study a special case of the deformed Virasoro algebra $\V_{p,q}$ with $q=-1$, to which we associate a Clifford vertex superalgebra and its $G$-covariant $\phi$-coordinated quasi modules.