Examples of $G/N=Q$ where $G$ and $Q$ are fixed, but $N$ have different choices.

Question

Let us say $G,N,Q$ are some groups (e.g. finite groups).

Are there any examples of $G/N=Q$ where $G$ and $Q$ are fixed and given, but $N$ have different choices?

Namely, say $G/N_1=Q$ and $G/N_2=Q$, with $G$ and $Q$ are fixed, but $N_1 \neq N_2.$

If there are no such examples, how can we prove all $N_1=N_2$?

Answer 1

Suppose $G = D_4$ is the symmetry group of the square.

$G$ has a normal subgroup $N_1$ that contains the rotational symmetries. $N_1$ is isomorphic to the cyclic group $C_4$.

$G$ has another normal subgroup $N_2$, which contains the identity, the 180-degree rotation, and the reflections about the two axes of the square. $N_2$ is isomorphic to Klein group $K_4$.

Yet $G/ N_1$ and $G/N_2$ are both isomorphic to the cyclic group $C_2$.