  A Kodaira fibration is a compact, complex surface admitting a holomorphic
submersion onto a complex curve, such that the fibers have nonconstant moduli.
We consider Kodaira fibrations X with nontrivial invariant rational cohomology
in degree 1, proving that if the dimension of the holomorphic invariants is 1
or 2, then X admits a branch covering over a product of curves inducing an
isomorphism on rational cohomology in degree 1. We also study the class of
Kodaira fibrations possessing a holomorphic section, and demonstrate that
having a section imposes no restriction on possible monodromies.
