In large dimensions, the only known compact, simply connected Riemannian manifolds with positive sectional curvature are spheres and projective spaces. The natural metrics on these spaces have large isometry groups, so it is natural to consider highly symmetric metrics when searching for new examples. On the other hand, there are many topological obstructions to a manifold admitting a positively curved metric with large symmetry. I will discuss a new obstruction in this setting. This is joint work with Manuel Amann (KIT).