  We show that the proportion of permutations $g$ in $S_n$ or $A_n$ such that
$g$ has even order and $g^{|g|/2}$ is an involution with support of cardinality
at most $\lceil n^\varepsilon \rceil$ is at least a constant multiple of
$\varepsilon$. Using this result, we obtain the same conclusion for elements in
a classical group of natural dimension $n$ in odd characteristic that have even
order and power up to an involution with $(-1)$-eigenspace of dimension at most
$\lceil n^\varepsilon \rceil$ for a linear or unitary group, or $2\lceil
\lfloor n/2 \rfloor^\varepsilon \rceil$ for a symplectic or orthogonal group.
