Homotopy complex projective spaces with Pin(2)-action

Let M be a manifold homotopy equivalent to the complex projective space CPm. Petrie conjectured that M has standard total Pontrjagin class if M admits a non-trivial action by S1. We prove the conjecture for m<12 under the assumption that the action extends to a nice Pin(2)-action with fixed point. The proof involves equivariant index theory for Spinc-manifolds and Jacobi functions as well as classical results from the theory of transformation groups.