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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 
$\lceil n^{{\it\varepsilon}}\rceil$ is at least a constant multiple of 
${\it\varepsilon}$. 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 
$\lceil n^{{\it\varepsilon}}\rceil$ for a linear or unitary group, or 
$2\lceil \lfloor n/2\rfloor ^{{\it\varepsilon}}\rceil$ for a symplectic or orthogonal group.
