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  ome  $i \in \{1,2,\ldots,6 \}$

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