  This paper continues with Part I. We define the module for a
$\frac{\mathbb{Z}}{2}$-graded meromorphic open-string vertex algebra that is
twisted by an involution and show that the axioms are sufficient to guarantee
the convergence of products and iterates of any number of vertex operators. A
module twisted by the parity involution is called a canonically
$\mathbb{Z}_2$-twisted module. As an example, we give a fermionic construction
of the canonically $\mathbb{Z}_2$-twisted module for the
$\frac{\mathbb{Z}}{2}$-graded meromorphic open-string vertex algebra
constructed in Part I. Similar to the situation in Part I, the example is also
built on a universal $\mathbb{Z}$-graded non-anti-commutative Fock space where
a creation operator and an annihilation operator satisfy the fermionic
anti-commutativity relation, while no relations exist among the creation
operators or among the zero modes. The Wick's theorem still holds, though the
actual vertex operator needs to be corrected from the na\"ive definition by
normal ordering using the $\exp(\Delta(x))$-operator in Part I.
