Let $G,H$ be finite groups. Suppose we have a epimorphism $G\times G\rightarrow H\times H$. Can we find an epimorphism $G\rightarrow H$?

A fellow graduate student asked me this question during TA sessions. Baffled, I asked this question on *mathstackexchange* [site][1], received some positive votes but no answer. According to him he has been running a software check on small order groups for days, and still have not find any counter example. So I venture to ask in here. It 'feels' unlikely to be true, yet we cannot find a proof or a counter example.

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subgroupbynormal subgrouphere: Let $H=A_4$ be the alternating group of order $12$. Then $(C_2\times C_2)\times 1$ is normal in $A_4\times A_4$, but $A_4$ has no normal subgroup of order $2$. – Peter Mueller Nov 7 '12 at 13:34