Let
G
G
be a semisimple simply connected algebraic group defined and split over the field
F
p
{\mathbb {F}}_p
with
p
p
elements, let
G
(
F
q
)
G(\mathbb {F}_{q})
be the finite Chevalley group consisting of the
F
q
{\mathbb {F}}_{q}
-rational points of
G
G
where
q
=
p
r
q = p^r
, and let
G
r
G_{r}
be the
r
r
th Frobenius kernel. The purpose of this paper is to relate extensions between modules in
Mod
(
G
(
F
q
)
)
\text {Mod}(G(\mathbb {F}_{q}))
and
Mod
(
G
r
)
\text {Mod}(G_{r})
with extensions between modules in
Mod
(
G
)
\text {Mod}(G)
. Among the results obtained are the following: for
r
>
2
r >2
and
p
≥
3
(
h
−
1
)
p\geq 3(h-1)
, the
G
(
F
q
)
G(\mathbb {F}_{q})
-extensions between two simple
G
(
F
q
)
G(\mathbb {F}_{q})
-modules are isomorphic to the
G
G
-extensions between two simple
p
r
p^r
-restricted
G
G
-modules with suitably “twisted" highest weights. For
p
≥
3
(
h
−
1
)
p \geq 3(h-1)
, we provide a complete characterization of
H
1
(
G
(
F
q
)
,
H
0
(
λ
)
)
\text {H}^{1}(G(\mathbb {F}_{q}),H^{0}(\lambda ))
where
H
0
(
λ
)
=
ind
B
G
λ
H^{0}(\lambda )=\text {ind}_{B}^{G}\ \lambda
and
λ
\lambda
is
p
r
p^r
-restricted. Furthermore, for
p
≥
3
(
h
−
1
)
p \geq 3(h-1)
, necessary and sufficient bounds on the size of the highest weight of a
G
G
-module
V
V
are given to insure that the restriction map
H
1
(
G
,
V
)
→
H
1
(
G
(
F
q
)
,
V
)
\operatorname {H}^{1}(G,V)\rightarrow \operatorname {H}^{1}(G(\mathbb {F}_{q}),V)
is an isomorphism. Finally, it is shown that the extensions between two simple
p
r
p^r
-restricted
G
G
-modules coincide in all three categories provided the highest weights are “close" together.