= Solution
The <elliptization theorem> says that a closed oriented <three-manifold> with finite <fundamental group> has spherical geometry. When the <fundamental group> is a <cyclic group>, it is a <lens space>. Consequently the candidates are $L(5,q)$ with $q\in\{1,2,3,4\}$, using the same surgery orientation convention throughout.
The <oriented classification of lens spaces> says that $L(p,q)$ and $L(p,q')$ are related by an orientation-preserving <homeomorphism> exactly when $q'\equiv q^{\pm1}\pmod p$. Since $1^{-1}=1$, $4^{-1}=4$, and $2^{-1}=3$ modulo five, the complete list is
$$
\boxed{L(5,1),\qquad L(5,2)\cong L(5,3),\qquad L(5,4).}
$$
These are three distinct oriented <homeomorphism> classes. Reversing orientation interchanges $L(5,1)$ and $L(5,4)$; the class containing $L(5,2)$ and $L(5,3)$ admits an orientation-reversing <homeomorphism>. The <elliptization theorem> and the stated <oriented classification of lens spaces> are the general classification results used here.
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