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.