Finite group with isomorphic normal subgroups and non-isomorphic quotients?

Solution 1:

Take $G = \mathbb{Z}_4 \times \mathbb{Z}_2$, $H$ generated by $(0,1)$, $K$ generated by $(2,0)$. Then $H \cong K \cong \mathbb{Z}_2$ but $G/H \cong \mathbb{Z}_4$ while $G/K \cong \mathbb{Z}_2 \times \mathbb{Z}_2$.

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Finite group with isomorphic normal subgroups and non-isomorphic quotients?

Solution 1:

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