The Kervaire manifolds are closed, oriented, almost framed PL manifolds in dimension 4k+2 with Arf invariant 1. In certain exceptional dimensions (2, 6, 14, 30, 62 and perhaps 126) there exist smooth Kervaire manifolds. The talk will be about the existence of free smooth or PL actions of finite groups on Kervaire manifolds, starting with the existence of free involutions. In contrast with a product of spheres $S^{2k+1} \times S^{2k+1}$, it turns out that the Kervaire manifolds have very restricted symmetry. This is joint work (in progress) with Diarmuid Crowley.