Homotopy Types of Suspended $4$-manifolds

Given a closed, smooth, connected, orientable $4$-manifold $M$, whose integral homology groups can have $2$-torsion, we determine the homotopy decomposition of the double suspension ${\mathrm{\Sigma }}^{2}M$ as wedge sums of some elementary ${\mathbf{A}}_3^3$-complexes, which are $2$-connected finite complexes of dimension at most $6$. Furthermore, we utilize the Postnikov square (or equivalently Pontryagin square) to find sufficient conditions for the homotopy decompositions of ${\mathrm{\Sigma }}^{2}M$ to desuspend to that of $\mathrm{\Sigma }M$.