Bound of automorphisms of fibred surfaces in positive characteristic

Let $f\text{\hspace{0.17em}⁣}:S\to B$ be a fibred surface over an algebraically closed field $k$ of characteristic $p\ge 5$, where $S$ is a minimal smooth projective surface of general type and $B$ is a smooth projective curve of genus $b\ge 2$. We prove that the group of fibration-preserving automorphisms of $f$ has order at most $15658(K_S^2{)}^4$. Furthermore, we provide an example to show that the exponent $4$ of the polynomial bound is sharp.