Surface fibrations with large equivariant automorphism group

For a relatively minimal surface fibration $f: Xto C$, the equivariant automorphism group of $f$ is, roughly speaking, the group of automorphisms of $X$ preserving the fibration structure. We present a classification of such fibrations of fibre genus $gge 1$ with smooth generic fibre over an arbitrary algebraically closed field $mathbf{k}$ whose equivariant automorphism group is infinite.