By definition, for coprime integers $p,q$, the lens space $L(p,q)$ is a quotient space of $S^3$ by the $\Bbb Z_p$-action generated by $(z,w)\mapsto (e^{2\pi i/p}z, e^{2\pi i q /p}w)$.
On the other hand, we can regard $S^3=SU(2)$ as a Lie group, and $S^3$ has cyclic subgroups of all orders. Is it true that, for any integer $p>1$ and any cyclic subgroup $G\subset S^3$ of order $p$, the quotient space $S^3/G$ is homeomorphic to $L(p,q)$ for some $q$? (Maybe this should be easy if I know all cyclic subgroups of $S^3$ of order $p$, which I don't know.)