In this question, which we flag it as a community wiki question, we search for a big list of groups $G$ which can not be isomorphic to a structure mentioned in $i.$ for some $i \in \{1,2,\ldots,6 \}$

1. The homeomorphism group of a topological space $X$.
2. The automorphism group of a (not necessarily finite dimensional) Lie algebra.
3. The automorphism group of a coalgebra.
4. (Assuming $G$ is a Lie group) The isometry group of a Riemannian manifold.
5. The automorphism group of another group.
6. The automorphism group of a ring.