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Let $G$ be a finite group $G$ and $\phi_1:G\to Aut(G)$ be a group homomorphism. Let $H_1=G\rtimes_{\phi_1} G$ (the (outer) semidirect product of $G$ and $G$ with respect to $\phi_1$). If there exists a group homomorphism $\phi_2:G\to Aut(H_1)$, then we can define $H_2=H_1\rtimes_{\phi_2} G$.

Is there some finite group $G$ for which we can define infinitely many $H_i$'s as the above which are not direct product?

Thanks so much for your helps

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Of course. You can always take $\varphi$ to be the trivial morphism for example. Then you get the sequence of direct products $G, G \times G, G \times G \times G \dots$

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  • $\begingroup$ Thanks so much. I always try to ask a question correctly, but again I always forget details and cause some problems. I edited my post and sorry so much $\endgroup$
    – khers
    Nov 12, 2018 at 9:29

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