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 (closed, orientable, connected) smooth $6$-manifold without torsion in $H^{3}$ has an almost complex structure hence a hope to be symplectic).