  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 $M(1)^{+}$.
