Let be a relatively minimal surface fibration with smooth generic fibre. We will discuss the finiteness of its equivariant automorphism group, which is the group of pairs with natural group law. We will give a complete classification of those surface fibrations with infinite equivariant automorphism group in any characteristic. As an application, we will show how this classification can be used to study the bounded subgroup property and the Jordan property for automorphism group of algebraic surfaces.