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Let $G = A * B$ be a central product of the finite groups $A$ and $B$. Suppose that $\operatorname{Aut}(A)$ and $\operatorname{Aut}(B)$ are solvable groups. Then is $\operatorname{Aut}(G)$ a solvable group?

And what about the same question with outer automorphism groups instead?

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    $\begingroup$ Where does $G$ come into play? $\endgroup$ – the_fox Nov 8 '18 at 14:16
  • $\begingroup$ Typo, fixed! 15chars $\endgroup$ – AnalysisStudent0414 Nov 8 '18 at 14:21
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In general no. A counterexample with the direct product to both questions is with $A$ and $B$ both cyclic of order $5$. ${\rm Aut}(A \times B) = {\rm GL}(2,5)$ is not solvable.

A counterexample with a genuine central product is $A=D_8$, $B=Q_8$, where again the outer automorphism group has $A_5$ as a composition factor.

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    $\begingroup$ Your first example is a genuine central product :) $\endgroup$ – YCor Nov 9 '18 at 8:52
  • $\begingroup$ Well I meant a central product that is not a direct product, but I expect you know that! $\endgroup$ – Derek Holt Nov 9 '18 at 10:28

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