Suppose that $A$ is a finitely generated abelian group without $2$-torsion. Does there exist a compact symplectic $6$-manifold $(M,\omega)$ such that $A$ Is isormorphic to $H_{3}(M,\mathbb{Z})$?

(The reason for excluding $2$-torsion is the fact that a $6$-manifold with torsion in $H_{3}$ has an almost complex structure hence a hope to be symplectic).