We study several geometric and group theoretical problems related to Kodaira fibrations, to more general families of Riemann surfaces, and to surface-by-surface groups. First we provide constraints on Kodaira fibrations that fiber in more than two distinct ways, addressing a question by Catanese and Salter about their existence. Then we show that if the fundamental group of a surface bundle over a surface is a CAT(0) group, the bundle must have injective monodromy (unless the monodromy has finite image). Finally, given a family of closed Riemann surfaces (of genus ≥2) with injective monodromy E→B over a manifold B, we explain how to build a new family of Riemann surfaces with injective monodromy whose base is a finite cover of the total space E and whose fibers have higher genus. We apply our construction to prove that the mapping class group of a once punctured surface virtually admits injective and irreducible morphisms into the mapping class group of a closed surface of higher genus.