Cohomotopy Sets of $(n-1)$-connected $(2n+2)$-manifolds for small $n$

Let $M$ be a closed orientable $(n-1)$-connected $(2n+2)$-manifold, $n\ge 2$. In this paper we combine the Postnikov tower of spheres and the homotopy decomposition of the reduced suspension space $\mathrm{\Sigma }M$ to investigate the cohomotopy sets ${\pi }^{\ast }(M)$ for $n=2,3,4$, under the assumption that $M$ has $2$-torsion-free homology. All cohomotopy sets ${\pi }^i(M)$ of such manifolds $M$ are characterized except ${\pi }^4(M)$ for $n=3,4$.