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

Quotients of $S^2\times{S^2}$