  We identify the group of homomorphisms
$\operatorname{Hom}_{\mathcal{GF}}(F,\mathbf{RU}_{\mathbb Q})$ in the category
of ($\operatorname{fin}$)-global functors to the rationalization of the unitary
representation ring functor and deduce that the higher
$\operatorname{Ext}$-groups
$\operatorname{Ext}^n_{\mathcal{GF}}(F,\mathbf{RU}_{\mathbb Q})$, $n\geq 2$
have to vanish. This leads to a rational splitting of the
($\operatorname{fin}$)-global equivariant $K$-theory spectrum into a sum of
Eilenberg-MacLane spectra. Interpreted in terms of cohomology theories, it
means that the equivariant Chern character is compatible with restrictions
along all group homomorphisms.
