Hodge structures of K3 type of bidouble covers of rational surfaces

A bidouble cover between complex algebraic varieties is a flat $G:=\left(\ZZ/2\ZZ\right)^2$-Galois cover $X \ra Y$. In this situation there exist three intermediate quotients $Y_1,Y_2$ and $Y_3$ which correspond to the three subgroups $\ZZ/2\ZZ \leq G$. We consider the following situation: $Y$ will be a rational surface and $Y_i$ will be either a surface with $p_g=0$ or a K3 surface. The motivation for these assumptions is to have a strong control on the weight 2 Hodge-structure of the covering surface $X$, which allows us to study the infinitesimal Torelli property, the Chow groups and Chow motive, and the Tate and Mumford-Tate conjectures for $X$. In particular, we classify all the bidouble covers $X\ra Y$ satisfing the conditions above whenever $Y$ is minimal; we obtain surfaces $X$ with $p_g(X)=1,2,3$.
We also introduce another construction, called \emph{iterated bidouble cover}, which allows us to obtain surfaces with higher value of $p_g$ for which we still have a strong control on the weight 2 Hodge-structure.