We consider closed topological 4-manifolds $M$ with universal cover
${S^2\times{S^2}}$ and Euler characteristic $\chi(M) = 1$. All such manifolds
with $\pi=\pi_1(M)\cong {\mathbb Z}/4$ are homotopy equivalent. In this case,
we show that there are four homeomorphism types, and propose a candidate for a
smooth example which is not homeomorphic to the geometric quotient. If
$\pi\cong {\mathbb Z}/2 \times {\mathbb Z}/2$, we show that there are three
homotopy types (and between 6 and 24 homeomorphism types).