Let
G
G
be a finite group. It is well known that a Mackey functor
{
H
↦
M
(
H
)
}
\{ H \mapsto M(H) \}
is a module over the Burnside ring functor
{
H
↦
Ω
(
H
)
}
\{ H \mapsto \Omega (H) \}
, where
H
H
ranges over the set of all subgroups of
G
G
. For a fixed homomorphism
w
:
G
→
{
−
1
,
1
}
w : G \to \{ -1, 1 \}
, the Wall group functor
{
H
↦
L
n
h
(
Z
[
H
]
,
w
|
H
)
}
\{ H \mapsto L_n^h ({\mathbb Z}[H], w|_H) \}
is not a Mackey functor if
w
w
is nontrivial. In this paper, we show that the Wall group functor is a module over the Burnside ring functor as well as over the Grothendieck-Witt ring functor
{
H
↦
G
W
0
(
Z
,
H
)
}
\{ H \mapsto {\mathrm {GW}}_0 ({\mathbb Z}, H) \}
. In fact, we prove a more general result, that the functor assigning the equivariant surgery obstruction group on manifolds with middle-dimensional singular sets to each subgroup of
G
G
is a module over the Burnside ring functor as well as over the special Grothendieck-Witt ring functor. As an application, we obtain a computable property of the functor described with an element in the Burnside ring.