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 $F_2$.