© 2016 Australian Mathematical Publishing Association Inc.We show that the proportion of permutations g in Sn or Ansuch that has even order and gg/2is an involution with support of cardinality at most [m?] is at least a constant multiple of . Using this result, we obtain the same conclusion for elements in a classical group of natural dimension in odd characteristic that have even order and power up to an involution with (-1)-eigenspace of dimension at most [m?] for a linear or unitary group, or 2[n/2?]for a symplectic or orthogonal group.