Are quotient groups unique up to isomorphism

Question

By this post, it seems quotient groups are unique up to isomorphism. is it correct? More clearly

Let $G$ be a group and let $K,N\unlhd G$ be isomorphic normal subgroups. Are $\frac{G}{N}$ and $\frac{G}{K}$ isomorphic?

Answer 1

$\mathbb{Z}/2\mathbb{Z}\neq\mathbb{Z}/3\mathbb{Z}$

Answer 2

Your statement is incorrect, but if the subgroups are isomorphic as subobjects (i.e. an isomorphism that commute with the inclusions) then it's true by general nonsense (i.e. category theory).