  We continue a previous study on $\Gamma$-vertex algebras and their
quasimodules. In this paper we refine certain known results and we prove that
for any $\Z$-graded vertex algebra $V$ and a positive integer $N$, the category
of $V$-modules is naturally isomorphic to the category of quasimodules of a
certain type for $V^{\otimes N}$. We also study certain generalizations of
twisted affine Lie algebras and we relate such Lie algebras to vertex algebras
and their quasimodules, in a way similar to that twisted affine Lie algebras
are related to vertex algebras and their twisted modules.
