The topological groups $String$ and $BS^1$ are - a priori - only defined up to homotopy equivalence. In that setting, it makes sense to talk about fibre sequences, but the question for a group extension, as well as the question of whether or not a group is abelian, depend on models.

In Stephan Stolz's original work about the string group [1], reviewed in [2], he realizes String as an extension of topological groups
$$
Gau(P) \to String \to Spin.
$$
The group $Gau(P)$ is the group of gauge transformations in a principal $PU(H)$-bundle $P$ over $Spin$, topologized in some appropriate way. It is normal in $String$, but it is not abelian. It also is not isomorphic as topological groups to $PU(H)$, $\mathbb{CP}^\infty$, or $BS^1$, but all these groups are weakly homotopy equivalent.

When working with (infinte-dimensional, strict) 2-groups and *strictly* exact sequences, the work of Baez et al. [3] gives a concrete model for a $String$-2-group, fitting into an exact sequence of 2-groups
$$
\mathcal{L}Spin \to String \to Spin
$$
(where $Spin$ is regarded as a 2-group with only identity morphisms). The kernel 2-group $\mathcal{L}Spin$ has objects $\Omega Spin$, the based loop group, and morphisms a semi-direct product of $\Omega Spin$ and its universal central extension. By geometric realization, one obtains another group extension of topological groups,
$$
|\mathcal{L}Spin| \to String \to Spin
$$
with some other group $\mathcal{L}Spin$ whose homotopy type is again a $K(\mathbb Z,2)$.

Allowing non-strict (homotopy-)exact sequences of strict 2-groups, any model of the string 2-group is a *central extension*
$$
BS^1 \to String \to Spin
$$
by the abelian 2-group $BS^1$. This is telling us (at least me) that this is a good setting. A very simple such model was described lately in [4].
Note that geometric realization will *not* result into an exact sequence of topological groups.

[1] *Stolz, Stephan*, **A conjecture concerning positive Ricci curvature and the Witten genus**, Math. Ann. 304, No. 4, 785-800 (1996). ZBL0856.53033.

[2] *Nikolaus, Thomas; Sachse, Christoph; Wockel, Christoph*, **A smooth model for the string group**, Int. Math. Res. Not. 2013, No. 16, 3678-3721 (2013). ZBL1339.22009.

[3] *Baez, John C.; Stevenson, Danny; Crans, Alissa S.; Schreiber, Urs*, **From loop groups to 2-groups**, Homology Homotopy Appl. 9, No. 2, 101-135 (2007). ZBL1122.22003.

[4] *Ludewig, Matthias; Waldorf, Konrad*, **Lie 2-groups from loop group extensions**