  Let $D(M,N)$ be the set of integers that can be realized as the degree of a
map between two closed connected orientable manifolds $M$ and $N$ of the same
dimension. For closed $3$-manifolds with $S^3$-geometry $M$ and $N$, every such
degree $deg f\equiv \overline{deg}\psi$ $(|\pi_1(N)|)$ where $0\le
\overline{deg}\psi <|\pi_1(N)|$ and $\overline{deg}\psi$ only depends on the
induced homomorphism $\psi=f_{\pi}$ on the fundamental group. In this paper, we
calculate explicitly the set $\{\overline{deg}\psi\}$ when $\psi$ is surjective
and then we show how to determine $\overline{deg}(\psi)$ for arbitrary
homomorphisms. This leads to the determination of the set $D(M,N)$.
