We study embeddings of
P
S
L
2
(
p
a
)
\mathrm {PSL}_2(p^a)
into exceptional groups
G
(
p
b
)
G(p^b)
for
G
=
F
4
,
E
6
,
2
E
6
,
E
7
G=F_4,E_6,{}^2\!E_6,E_7
, and
p
p
a prime with
a
,
b
a,b
positive integers. With a few possible exceptions, we prove that any almost simple group with socle
P
S
L
2
(
p
a
)
\mathrm {PSL}_2(p^a)
, that is maximal inside an almost simple exceptional group of Lie type
F
4
F_4
,
E
6
E_6
,
2
E
6
{}^2\!E_6
and
E
7
E_7
, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup of type
A
1
A_1
inside the algebraic group.
Together with a recent result of Burness and Testerman for
p
p
the Coxeter number plus one, this proves that all maximal subgroups with socle
P
S
L
2
(
p
a
)
\mathrm {PSL}_2(p^a)
inside these finite almost simple groups are known, with three possible exceptions (
p
a
=
7
,
8
,
25
p^a=7,8,25
for
E
7
E_7
).
In the three remaining cases we provide considerable information about a potential maximal subgroup.