At least for compact groups, I think your question was answered by Peterson and Wilhelm where they showed the Gromoll-Meyer sphere has a metric of positive sectional curvature.
That, combined with work of Wu-Chung Hsiang's work from the 60's says that exotic spheres are not homogeneous spaces. I'm not sure which is the best reference but Wu-Chung has a few on similar topics, for example, his paper "On Compact Subgroups of Diffeomorphism Groups of Kervaire Spheres" Annals 1967 mentions that compact subgroups of Diff of an exotic sphere has dimension bounded above by $m^2/8 + 1$, where $m$ is the dimension of the sphere.