Normal automorphisms of free Burnside groups

We prove that for an arbitrary odd $n\geqslant1003$ and $m>1$ every automorphism of the free Burnside group $B(m,n)$ that stabilizes every maximal normal subgroup $N\trianglelefteq B(m,n)$ of infinite index is an inner automorphism. For the same values of $m$ and $n$, we establish that the subgroup of inner automorphisms of $\operatorname{Aut}(B(m,n))$ is maximal among the subgroups in which the orders of the elements are bounded by $n$.