Are there groups $G$ and $H$ such that $G$ and $H$ are not isomorphic but $G \times \mathbb Z$ and $H \times \mathbb Z$ are?
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Yes, there is an example in a paper by Hirshon, "On Cancellation in Groups". You can find a slightly different example written up here
Note that if one of them is abelian and finitely generated then by structure theorem $G$ and $H$ are isomorphic.