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I am studying for an upcoming exam and was given the following problem as practice:

Can it happen that $G_1$ is not isomorphic to $G_2$ yet have isomorphic normal subgroups $N_1 < G_1$ and $N_2 < G_2$ and isomorphic quotient groups $G_1/N_1$ and $G_2/N_2$?

I think it must be that $G_1$ is isomorphic to $G_2$ but I cannot for the life of me show this.

One situation might be that the order of $G_1$ and $G_2$ are not the same then the order of $G_1/N_1$ is not the same as $G_2/N_2$ but if $|G_1|=|G_2|$ it seems like its possible for them to not be isomorphic yet display the above properties.

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Consider $\mathbb Z/2 \oplus \mathbb Z/2$ and $\mathbb Z/4$.

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