Maximal actions of finite 2-groups on $Bbb{Z}_2$ - homology 3-spheres

It is known that a finite 2-group acting on a Z_2-homology 3-sphere has at most ten conjugacy classes of involutions; the action of groups with the maximal number of conjugacy classes of involutions is strictly related to some questions concerning the representation of hyperbolic 3-manifolds as 2-fold branched coverings of knots. Using a low-dimensional approach we classify these maximal actions both from an algebraic and from a geometrical point of view.