What is more interesting is to look at the *fundamental groupoid* $\pi_1 G$ of a topological group, particularly in the non-connected case. This groupoid inherits a group structure, and so becomes what is called a group-groupoid, or a $2$-group, i.e. a group internal to the category of groupoids. Such objects are equivalent to *crossed modules*, which are objects which model connected, pointed weak homotopy 2-types.

Here are a couple of relevant papers:

R. Brown and C.B. Spencer, $\cal G$-groupoids, crossed modules
and the fundamental groupoid of a topological group'', *Proc.
Kon. Ned. Akad. v. Wet.* 7 (1976) 296-302.

R. Brown and O. Mucuk, ``Covering groups of non-connected
topological groups revisited'', *Math. Proc. Camb. Phil.
Soc*, 115 (1994) 97-110.

and just now a search on Baez 2-groups gave 82,700 hits, e.g.

Higher-Dimensional Algebra V: 2-Groups. John C. Baez.

The second paper revisits the work of R.L. Taylor which showed that there is in general an obstruction to a non-connected topological group having a universal covering topological group.

Later: The following may be seen as still not answering the original question, but I hope will be interesting to some readers!

I was trying to convey that the fundamental groupoid $\Phi=\pi_1 G$ of a topological group $G$ contains useful information, so I hope it will be useful to set that out in some, but not full, detail.

As said above, $\Phi$ is in fact a group-groupoid, and so has an associated crossed module, from which may be recovered the group-groupoid. This crossed module, say $\delta: C \to G$, is part, see below, of an exact sequence

$$ 0 \to \pi_1(G,e) \to C \to G \to \pi_0 G \to 1$$

which is known as a *crossed sequence*. This crossed sequence determines a cohomology class $ k \in H^3(\pi_0 G, \pi_1 (G,e)) $, as shown by Mac Lane, which may also be identified with the first Postnikov invariant of the classifying space $BG$. This invariant is trivial if and only if $G$ admits a universal covering group (assuming $G$ is suitably nice locally), by which is meant a topological group $U$ with a covering map $p: U \to G$ which is also a homomorphism of topological groups, and such that $p$ restricts to a universal covering map for each component of $G$.

Now to give more on the crossed module $\delta: S \to G$. According to one convention, $C$ is the costar (or star in another convention) of $\pi_1 G$ at $e$, i.e. the elements of $\pi_1 G$ which end at $e$. The map $\delta$ then is just the source map. The group structure on $C$ is induced by the multiplication in $G$: i.e. on path classes $[a][b]=[c]$ where $c(t)=a(t)b(t)$. The operation of $G$ on $C$ is by conjugation: $[a]^g= [g^{-1}ag]$. The crossed module rules are of the form: CM1) $\delta(\alpha ^g)= g^{-1}(\delta \alpha)g $; CM2) $\alpha^{-1} \beta \alpha = \beta ^{\delta \alpha}$, for all $\alpha, \beta \in C, g \in G$.

Notice that in the crossed sequence displayed, the fundamental group is the kernel of $\delta$, and is an abelian group and in fact a module over $\pi_0 G$. But this abelian group, and even the module, is, in the non-connected case, and in general, but a pale shadow of the $2$-type of the classifying space.

What I have been trying to convey is the idea that throwing away a larger structure, in this case restricting from a groupoid to a group, may throw away needed information. Thus in homotopy theory there is a concentration on homotopy groups, e.g. second homotopy groups. But it has been shown that in some cases the best way to determine this group may be, indeed may only be, by calculating the whole $2$-type, as determined by a crossed module. This seems to me an interesting inversion of a traditional approach.

Maybe this site is not the place for such remarks?

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