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

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 $\pi ={\pi }_1(M)\cong \mathbb{Z}\mathrm{/}4$ 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 $\pi \cong \mathbb{Z}\mathrm{/}2\times \mathbb{Z}\mathrm{/}2$, we show that there are three homotopy types (and between 6 and 24 homeomorphism types).