  To give a unified treatment on the association of Lie algebras and vertex
algebras, we study $(G,\chi_\phi)$-equivariant $\phi$-coordinated quasi modules
for vertex algebras, where $G$ is a group with $\chi_\phi$ a linear character
of $G$ and $\phi$ is an associate of the one-dimensional additive formal group.
The theory of $(G,\chi_\phi)$-equivariant $\phi$-coordinated quasi modules for
nonlocal vertex algebra is established in \cite{JKLT}. In this paper, we
concentrate on the context of vertex algebras. We establish several conceptual
results, including a generalized commutator formula and a general construction
of vertex algebras and their $(G,\chi_\phi)$-equivariant $\phi$-coordinated
quasi modules. Furthermore, for any conformal algebra $\mathcal{C}$, we
construct a class of Lie algebras $\widehat{\mathcal{C}}_\phi[G]$ and prove
that restricted $\widehat{\mathcal{C}}_\phi[G]$-modules are exactly
$(G,\chi_\phi)$-equivariant $\phi$-coordinated quasi modules for the universal
enveloping vertex algebra of $\mathcal{C}$. As an application, we determine the
$(G,\chi_\phi)$-equivariant $\phi$-coordinated quasi modules for affine and
Virasoro vertex algebras.
