Splitting automorphisms of free Burnside groups

It is proved that, if the order of a splitting automorphism of odd period $n\geqslant1003$ of a free Burnside group $B(m,n)$ is a prime, then the automorphism is inner. This implies, for every prime $n\geqslant1009$, an affirmative answer to the question on the coincidence of the splitting automorphisms of period $n$ of the group $B(m,n)$ with the inner automorphisms (this question was posed in the “Kourovka Notebook” in 1990).
Bibliography: 17 titles.