  The involution fixity ${\rm ifix}(G)$ of a permutation group $G$ of degree
$n$ is the maximum number of fixed points of an involution. In this paper we
study the involution fixity of primitive almost simple exceptional groups of
Lie type. We show that if $T$ is the socle of such a group, then either ${\rm
ifix}(T) > n^{1/3}$, or ${\rm ifix}(T) = 1$ and $T = {}^2B_2(q)$ is a Suzuki
group in its natural $2$-transitive action of degree $n=q^2+1$. This bound is
best possible and we present more detailed results for each family of
exceptional groups, which allows us to determine the groups with ${\rm ifix}(T)
\leqslant n^{4/9}$. This extends recent work of Liebeck and Shalev, who
established the bound ${\rm ifix}(T) > n^{1/6}$ for every almost simple
primitive group of degree $n$ with socle $T$ (with a prescribed list of
exceptions). Finally, by combining our results with the Lang-Weil estimates
from algebraic geometry, we determine bounds on a natural analogue of
involution fixity for primitive actions of exceptional algebraic groups over
algebraically closed fields.
