Finite group actions on Kervaire manifolds

Let MK4k+2 be the Kervaire manifold: a closed, piecewise linear (PL) manifold with Kervaire invariant 1 and the same homology as the product S2k+1×S2k+1 of spheres. We show that a finite group of odd order acts freely on MK4k+2 if and only if it acts freely on S2k+1×S2k+1. If MK is smoothable, then each smooth structure on MK admits a free smooth involution. If k≠2j−1, then MK4k+2 does not admit any free TOP involutions. Free “exotic” (PL) involutions are constructed on MK30, MK62, and MK126. Each smooth structure on MK30 admits a free Z/2×Z/2 action.