Can we have some examples of group extensions $G/N=Q$ with continuous (e.g. topological groups or Lie groups) $G$ and $N$, but a finite discrete $Q$? Note that $1 \to N \to G \to Q \to 1$.
What else are the examples that you can provide?
A systematic answer to obtain new examples (a few or even a list) is of course most welcome. ; )
Another related questions: What I know already knew are some examples of continuous (e.g. Lie groups) $G$ and $Q$, but a finite discrete $N$ here (but I am looking for more examples there, too --- because the examples there are only two).
Just choose an open, normal subgroup of a compact group, eg.
So long as the upstairs group is Hausdorff, the open subgroups are exactly the closed subgroups of finite index, so you pick any abelian group and you're good to go.
You can do similarly with $p$-adic units, eg
$$1\to 1+p\Bbb Z_p\to \Bbb Z_p^\times\to (\Bbb Z/p)^\times\to 1$$
or
$$1\to 1+p^2\Bbb Z_p\to 1+p\Bbb Z_p\to \Bbb Z/p\to 1$$
or if you prefer something a bit more exotic, do the same with polynomial or power series rings eg.
$$1\to \langle p(t)\rangle\to \Bbb F_p[t]\to \Bbb F_p(\alpha)\to 1$$
for any irreducible $p(t)\in \Bbb F_p[t]$ and $\alpha$ a root.