I've been searching for an answer for weeks, so either I'm bad at searching or this question has not been asked yet. What I'm interested in is basically the title of the question: do generalizations of lens spaces $S^3/\mathbb{Z}_n$ exist for the Hopf sphere $S^7$? That is $S^7/\Gamma$ where $\Gamma$ is a finite subgroup of $SU(2)$, that is either a cyclic, dihedral, tetrahedral, octahedral, or icosahedral group. (I know they exist for $\Gamma=\mathbb{Z}_n$, I'm mostly interested in the other cases.)
I read various papers of Boyer and Galicki on 3-Sasakian manifolds but I could not find an answer to this question. I suspect the answer to the question is "yes", but I would appreciate if someone could provide some reference.
In case of positive answer I would be interested in the following. I understand that the large $n$ limit of $S^3/Z_n$ is $S^2$ (where $\mathbb{Z}_n$ acts as $(z_1,z_2)\sim(\lambda z_1,\lambda z_2)$ and $\lambda=\exp(2\pi i/n)$) as it identifies all points in the Hopf fiber. Is there a finite subgroup $\Gamma$ of $SU(2)$ such that taking the order of $\Gamma$ to be large the quotient $S^7/\Gamma$ gives $S^4$? That is it identifies all points in the $S^3$ Hopf fiber. (It clearly cannot be $\Gamma=\mathbb{Z}_n$.)