Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 3 / 1D / iii / Solution

False. Let $G=C_4$ and $H=C_2\times C_2$, and in each choose a subgroup of order two. Then the chosen normal subgroups are isomorphic and both quotients are $C_2$, while $G$ and $H$ are not isomorphic.