On extension of a regular automorphism (Q6175225)

The paper discusses the following problem asked by Zhurtov: Question. Let \(G\) be a periodic group, let \(A\) be an abelian normal subgroup of \(G\) coinciding with its centralizer, and let \(t\) be an automorphism of prime order of \(A\) acting on \(A\) without non-trivial fixed points and centralizing the quotient group \(G/A\) considered as the automorphism group of \(A\). Is it true that \(t\) can be extended to the automorphism of \(G\)? In the full generality, the answer is negative, indeed the authors report first a counterexample constructed by Olshanskii. However, they are able to obtain a positive answer under the additional condition that \(\pi(A)\) and \(\pi(G/A)\) are disjoint where \(\pi(X)\) denotes the set of all prime divisors of element orders of a periodic group \(X\). Indeed, they prove the following nice result: Theorem. Suppose that \(A\) is an abelian normal subgroup of a periodic group \(G\) which coincides with its centralizer in \(G\), and \(t\) is an automorphism of prime order of \(A\) without nontrivial fixed points. If \(\pi(A) \cap \pi(G/A) = \emptyset\) and \(t\) centralizes the quotient group \(G/A\), considered as a subgroup of the automorphism group of \(A\), then \(t\) can be extended to an automorphism \(\tau\) of \(G\). Moreover, \(C_{G}(\tau)\), the group of fixed points of the automorphism \(\tau\) in \(G\), is a complement to \(A\) in \(G\).