How large is the braid monodromy of low-genus Lefschetz fibrations?

Given a genus $g$ smooth Lefschetz fibration $\pi :M\to S^2$ with singular locus $\mathrm{\Delta }\subseteq S^2$, we describe the subgroup $\mathrm{Br}(\pi )$ of the spherical braid group $\mathrm{Mod}(S^2,\mathrm{\Delta })$ consisting of braids admitting a lift to a fiber-preserving diffeomorphism of $M$. We develop general methods for showing that the index $[\mathrm{Mod}(S^2,\mathrm{\Delta }):\mathrm{Br}(\pi )]$ is infinite. As an application of our methods, we prove that $[\mathrm{Mod}(S^2,\mathrm{\Delta }):\mathrm{Br}(\pi )]=\mathrm{\infty }$ when $g=1$, when $\pi$ is expressible as a self-fiber sum when $g\ge 2$, or when $\pi$ is a holomorphic genus $g=2$ Lefschetz fibration whose vanishing cycles are nonseparating. In the genus $g=1$ case, we relate the subgroup $\mathrm{Br}(\pi )$ to the action of $\mathrm{Mod}(S^2,\mathrm{\Delta })$ on the ${\mathrm{SL}}_2$-character variety for $S^2\setminus \mathrm{\Delta }$ and provide an alternate proof of the first application via recent work of Lam–Landesman–Litt.