For \(0\leq k\leq 1\), the finite group G is called k-group if \(k| G|\) is the greatest number of elements of G inverted by any automorphism of G. k-groups with \(k>1/2\) were classified by Liebeck and MacHale, and the present paper is concerned with the case \(k=1/2\). In the preliminary results, it is shown that if \(k\geq 1/2\), any Sylow p-subgroup for \(p\geq 5\) is normal and Abelian, and for \(k>1/2\) G is 2-nilpotent. It is also shown that if \(G=H\times K\), G is a 1/2-group if and only if either one factor is Abelian and the other a 1/2-group, or one factor is a 2/3-group and the other is a 3/4-group. So it can be assumed that the 1/2-group G has no Abelian direct fctor. If G is not 2-nilpotent, then G/Z(G) is isomorphic to the alternating group of degree 4 and G is a \(\{\) 2,3\(\}\)-group. If G is 2-nilpotent, then again G is a \(\{\) 2,3\(\}\)-group, and if G is not a 2-group, G/Z(G) is the direct product of two cyclic groups of order 2 and the symmetric group of degree 3. As a corollary, it is shown that if the group G of even order contains precisely 1/2\(| G| -1\) involutions, then G is either a 2-group or the direct product of the dihedral group of order 8, the symmetric group of degree 3 and an elementary Abelian 2-group.