Automorphisms of braid groups on orientable surfaces

In this paper we compute the automorphism groups $\mathrm{Aut}({\mathbf{P}}_n(\mathrm{\Sigma }))$ and $\mathrm{Aut}({\mathbf{B}}_n(\mathrm{\Sigma }))$ of braid groups ${\mathbf{P}}_n(\mathrm{\Sigma })$ and ${\mathbf{B}}_n(\mathrm{\Sigma })$ on every orientable surface $\mathrm{\Sigma }$, which are isomorphic to group extensions of the extended mapping class group ${\mathcal{M}}_n^{\ast }(\mathrm{\Sigma })$ by the transvection subgroup except for a few cases. We also prove that ${\mathbf{P}}_n(\mathrm{\Sigma })$ is always a characteristic subgroup of ${\mathbf{B}}_n(\mathrm{\Sigma })$ unless $\mathrm{\Sigma }$ is a twice-punctured sphere and $n=2$.