In this question, which we flag it as a community wiki quetion, we search for a big list of groups $G$ which can not be isomorphic to any one of the following structures: That is, a group $G$ which is not iomorphic to an structure mentioned in $i)$ for some $i \in \{1,2,\ldots,6 \}$

1. The homemorphism group of a topological space $X$.
2. The automorphism group of a (not necearilly 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.