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 $M(1)^{ +}$.
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