$K(2)$-local splittings of finite Galois extensions of $MU⟨6⟩$ and $MString$

Using a Milnor-Moore argument we show that, $K(2)$-locally at the prime $2$, the spectra $MU⟨6⟩$ and $MString$ split as direct sums of Morava $E$-theories after tensoring with a finite Galois extension of the sphere called $E^{h{F}_{3\mathrm{/}2}}$. In the case of $MString$ we are able to refine this splitting in several ways: we show that the projection maps are determined by spin characteristic classes, that the Ando-Hopkins-Rezk orientation admits a unital section after tensoring with $E^{h{F}_{3\mathrm{/}2}}$, and that the splitting can be improved to one of $E^{hH}\otimes MString$ into a direct sum of shifts of $TM{F}_0(3)$ where $H$ is an open subgroup of the Morava stabilizer group of index $4$.