  The (4k+2)-dimensional Kervaire manifold is a closed, piecewise linear (PL)
manifold with Kervaire invariant 1 and the same homology as the product of two
(2k+1)-dimensional spheres. We show that a finite group of odd order acts
freely on a Kervaire manifold if and only if it acts freely on the
corresponding product of spheres. If the Kervaire manifold M is smoothable,
then each smooth structure on M admits a free smooth involution. If k + 1 is
not a 2-power, then the Kervaire manifold in dimension 4k+2 does not admit any
free TOP involutions. Free "exotic" (PL) involutions are constructed on the
Kervaire manifolds of dimensions 30, 62, and 126. Each smooth structure on the
30-dimensional Kervaire manifold admits a free Z/2 x Z/2 action.
