  We study embeddings of $\mathrm{PSL}_2(p^a)$ into exceptional groups $G(p^b)$
for $G=F_4,E_6,{}^2\!E_6,E_7$, and $p$ a prime with $a,b$ positive integers.
With a few possible exceptions, we prove that any almost simple group with
socle $\mathrm{PSL}_2(p^a)$, that is maximal inside an almost simple
exceptional group of Lie type $F_4$, $E_6$, ${}^2\!E_6$ and $E_7$, is the fixed
points under the Frobenius map of a corresponding maximal closed subgroup of
type $A_1$ inside the algebraic group.
  Together with a recent result of Burness and Testerman for $p$ the Coxeter
number plus one, this proves that all maximal subgroups with socle
$\mathrm{PSL}_2(p^a)$ inside these finite almost simple groups are known, with
three possible exceptions ($p^a=7,8,25$ for $E_7$).
  In the three remaining cases we provide considerable information about a
potential maximal subgroup.
