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Then there is a finite collection $\\mathcal{F}$ of manifolds of the form $\\mathbb{S}^3 \\times \\mathbb{R} /G$, where $G$ is a discrete subgroup of the isometry group of the round cylinder $\\mathbb{S}^3\\times \\mathbb{R}$ on which $G$ acts freely, such that $X$ is diffeomorphic to a possibly infinite connected sum of $\\mathbb{S}^4,\\mathbb{RP}^4$ and members of $\\mathcal{F}$. This extends recent work of Chen-Tang-Zhu and Huang. 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