Cofiniteness and $P(z)$-tensor product bifunctors in orbifold theories associated to abelian but not-necessarily-finite groups

Let $V$ be a Möbius vertex algebra and $G$ an abelian group of automorphisms of $V$. We construct $P(z)$-tensor product bifunctors for the category of $C_n$-cofinite grading-restricted generalized $g$-twisted $V$-modules (without $g$-actions) for $g\in G$ and the category of $C_n$-cofinite grading-restricted generalized $g$-twisted $V$-modules with $G$-actions for $g\in G$. In this paper, an automorphism $g$ of $V$ can be of infinite order and does not have to act semisimply on $V$, and the group $G$ can be an infinite abelian group containing nonsemisimple automorphisms of $V$.