A unitary representation $\pi$ of a, possibly infinite dimensional, Lie group $G$ is called semibounded if the corresponding operators $i\mathtt{d}\pi \left(x\right)$ from the derived representation are uniformly bounded from above on some non-empty open subset of the Lie algebra $𝔤$ of $G$. We classify all irreducible semibounded representations of the groups ${\widehat{ℒ}}_{\phi }\left(K\right)$ which are double extensions of the twisted loop group ${ℒ}_{\phi }\left(K\right)$, where $K$ is a simple Hilbert–Lie group (in the sense that the scalar product on its Lie algebra is invariant) and $\phi$ is...