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:

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