Automorphism group of a topological space
Solution 1:
As pointed out in the comments, this has been answered by Tony Huynh on MathOverflow. In
de Groot, J. ($1959$), Groups represented by homeomorphism groups, Mathematische Annalen $138$
the author shows that:
"for every group $G$ one can find a complete, connected, locally connected metric space $M$ of any positive dimension such that $G \cong A(M)$"
where $A(M)$ denotes the autohomeomorphism group of $M$.