Surface fibrations with large equivariant automorphism group

For a relatively minimal surface fibration $f: X\to C$, 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 $g\ge 1$ with smooth generic fibre over an arbitrary algebraically closed field $\mathbf{k}$ whose equivariant automorphism group is infinite.