Automorphisms of group extensions. (Q2856431)

Every group extension \(\mathbf e:N\rightarrowtail G\twoheadrightarrow Q\) gives rise to homomorphism \(\chi\colon Q\to\text{Out}(N)\), called the coupling of the extension \(\mathbf e\), from \(Q\) to the group \(\text{Out}(N)\) of outer automorphisms of \(N\). It turns out that equivalent extensions have the same coupling. An extensively investigated problem in the theory of groups is to determine when a pair \((\alpha,\gamma)\in\Aut(N)\times\Aut(Q)\) of automorphisms is induced by some automorphism of the extension \(\mathbf e\) [see e.g., \textit{C. Wells}, Trans. Am. Math. Soc. 155, 189-194 (1971; Zbl 0221.20054); \textit{K.-W. Yang}, Pac. J. Math. 50, 299-304 (1974; Zbl 0251.20037); \textit{W. Malfait}, Bull. Belg. Math. Soc. - Simon Stevin 9, No. 3, 361-372 (2002; Zbl 1041.20018); \textit{P. Jin}, J. Algebra 312, No. 2, 562-569 (2007; Zbl 1131.20037); \textit{I. B. S. Passi, M. Singh} and \textit{M. K. Yadav}, J. Algebra 324, No. 4, 820-830 (2010; Zbl 1209.20021)].NEWLINENEWLINE The two main results of this paper are reduction theorems for the inducibility problem. In case \(Q\) is finite, Theorem 2 reduces the problem to the case of prime-power groups, and Theorem 3 addresses the case when \(Q\) is locally finite.