We investigate the interplay between properties of Ext modules and the ascent of module structures along local ring homomorphisms. Specifically, let
φ
:
(
R
,
m
,
k
)
→
(
S
,
m
S
,
k
)
\varphi \colon (R,\mathfrak {m},k)\to (S,\mathfrak {m} S,k)
be a flat local ring homomorphism. We show that if
M
M
is a finitely generated
R
R
-module such that
Ext
R
i
(
S
,
M
)
\operatorname {Ext}_{R}^{i}(S,M)
satisfies NAK (e.g. if
Ext
R
i
(
S
,
M
)
\operatorname {Ext}_{R}^{i}(S,M)
is finitely generated over
S
S
) for
i
=
1
,
…
,
dim
R
(
M
)
i=1,\ldots ,\dim _{R}(M)
, then
Ext
R
i
(
S
,
M
)
=
0
\operatorname {Ext}_{R}^{i}(S,M)=0
for all
i
≠
0
i\neq 0
and
M
M
has an
S
S
-module structure that is compatible with its
R
R
-module structure via
φ
\varphi
. We provide explicit computations of
Ext
R
i
(
S
,
M
)
\operatorname {Ext}_{R}^{i}(S,M)
to indicate how large it can be when
M
M
does not have a compatible
S
S
-module structure.