Splitting Automorphisms of Order $p^k$ of Free Burnside Groups are Inner

It is proved that, if the order of a splitting automorphism of odd period $n\ge 1003$ of a free Burnside group $B(m,n)$ is equal to a power of some prime, then the automorphism is inner. Thus, an affirmative answer is given to the question concerning the coincidence of the splitting automorphisms of the group $B(m,n)$ with the inner automorphisms for all automorphisms of order $p^k$ ($p$ is a prime). This question was posed in 1990 in “Kourovka Notebook” (see the 11th edition, Question 11.36.b).