Is there an example of a finite Galois extension $E/F$ of number fields, such that $G=\mathrm{Gal}(E/F)$ is non-abelian and the order of the cohomology group $H^1(G,U_E)$ is relatively prime to class number $N$? ($U_E$ denotes the group of units of $E$.)

(Indeed, I think if $E/F$ is an abelian (or at least is a cyclic) unramified (at all finite places) extension, it is not possible for $\# H^1(G,U_E)$ to be relatively prime to the class number $N$. Is that true?)