Let $H_1,H_2$ be finite 2-generated perfect groups. Does there exist a finite 2-generated perfect group $G$ which has $H_1,H_2$ as quotients?

Of course we may assume $H_1\not\cong H_2$. If $H_1,H_2$ are simple, then this is true, since any surjections $p_i : F_2\rightarrow H_i$ where $F_2$ is the free group of rank 2, induces a map $(p_1\times p_2) : F_2\rightarrow H_1\times H_2$ which is also a surjection by a Goursat's lemma argument, and direct products of perfect groups are perfect. If $H_1,H_2$ are not simple, then it's unclear if the image of $p_1\times p_2$ is still perfect.

This came up while I was thinking about the "pro-perfect completion" of $F_2$.

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    $\begingroup$ I would guess that the answer is no, but I haven't managed to find an example yet! $\endgroup$ – Derek Holt Nov 19 '15 at 16:00
  • $\begingroup$ @DerekHolt Do you know if there should be injections of $H_1,H_2$ into the image of $p_1\times p_2$? $\endgroup$ – oxeimon Nov 19 '15 at 19:40
  • $\begingroup$ I think I can find exampes where there are no such injections, but that doesn't answer the question you asked. I think there ought to be examples where the image is not perfect, but I haven't found any yet. $\endgroup$ – Derek Holt Nov 19 '15 at 22:18
  • $\begingroup$ @DerekHolt Would you happen to know anything or have any references about the pro-$\Delta$ completion of $F_2$? Here $\Delta$ is the class of finite groups whose simple composition factors are all nonabelian. $\endgroup$ – oxeimon Nov 20 '15 at 20:19

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