The irreducible weak modules for the fixed point subalgebra of the vertex algebra associated to a non-degenerate even lattice by an automorphism of order $2$ (Part $2$)

Let $V_L$ be the vertex algebra associated to a non-degenerate even lattice $L$, $\theta$ the automorphism of $V_L$ induced from the $-1$ symmetry of $L$, and $V_L^{+}$ the fixed point subalgebra of $V_L$ under the action of $\theta$. In this series of papers, we classify the irreducible weak $V_L^{+}$-modules and show that any irreducible weak $V_L^{+}$-module is isomorphic to a weak submodule of some irreducible weak $V_L$-module or to a submodule of some irreducible $\theta$-twisted $V_L$-module. Let $M(1{)}^{+}$ be the fixed point subalgebra of the Heisenberg vertex operator algebra $M(1)$ under the action of $\theta$. In this paper (Part $2$), we show that there exists an irreducible $M(1{)}^{+}$-submodule in any non-zero weak $V_L^{+}$-module and we compute extension groups for $M(1{)}^{+}$.