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Let 
$G$ be a finite permutation group of degree 
$n$ and let 
${\rm ifix}(G)$ be the involution fixity of 
$G$, which is the maximum number of fixed points of an involution. In this paper, we study the involution fixity of almost simple primitive groups whose socle 
$T$ is an alternating or sporadic group; our main result classifies the groups of this form with 
${\rm ifix}(T) \leqslant n^{4/9}$. This builds on earlier work of Burness and Thomas, who studied the case where 
$T$ is an exceptional group of Lie type, and it strengthens the bound 
${\rm ifix}(T) > n^{1/6}$ (with prescribed exceptions), which was proved by Liebeck and Shalev in 2015. A similar result for classical groups will be established in a sequel.
