Surface fibrations with large equivariant automorphism groups

Given a fibration $f: X\to C$ from a smooth projective surface $X$ to a smooth curve $C$ over an arbitrary algebraically closed field $k$. The equivariant automorphism group is
$$\mathbb{E}(X/C):=\{\,\,(\tau,\sigma) \,\,|\,\, \tau\in \mathrm{Aut}_k(X), \sigma\in \mathrm{Aut}_k(C),\,\,\, f\circ \tau =\sigma\circ f\,\, \}$$
with natural composition law.
$$ \xymatrix{ X\ar[rr]_\sim^\tau \ar[d]_f&& X \ar[d]^f\\ C \ar[rr]_\sim^\sigma && C } $$
This group is an important invariant of the fibration $f$ and sometimes that of the surface $X$. In this talk, we will give a classification of those relatively minimal surface fibrations whose equivariant automorphism group $\mathbb{E}(X/C)$ is infinite. As an application, we will also discuss the Jordan property of the automorphism group of a minimal surface of Kodaira dimension one.