I am currently studying mapping class groups. In particular, I am looking at a relation between the group of topological automorphisms of a topological group (i.e. group automorphisms which are also homeomorphisms of the underlying space) and the mapping class group of the underlying space. Denote the topological automorphism group of a topological group $G$ as $TI(G)$. It is clear that we should look at the mapping class group of the underlying space with a marked point corresponding to the identity. It is the case that if $G$ is $K(\pi,1)$ for some abelian group $\pi$ (since the fundamental group of a topological group is always abelian) then by the Dehn-Nielsen-Baer theorem $Mod(G)\cong Aut(\pi)$. However, I am uncertain as to whether there are many topologcial groups which are in fact Eilenberg-Maclane spaces. So in other words, are there constructions of more complicated topological groups from ones which are Eilenberg-Maclane spaces?