Let
V
V
be a vertex algebra of countable dimension,
G
G
a subgroup of
A
u
t
V
AutV
of finite order,
V
G
V^{G}
the fixed point subalgebra of
V
V
under the action of
G
G
, and
S
\mathscr {S}
a finite
G
G
-stable set of inequivalent irreducible twisted weak
V
V
-modules associated with possibly different automorphisms in
G
G
. We show a Schur–Weyl type duality for the actions of
A
α
(
G
,
S
)
\mathscr {A}_{\alpha }(G,\mathscr {S})
and
V
G
V^G
on the direct sum of twisted weak
V
V
-modules in
S
\mathscr {S}
where
A
α
(
G
,
S
)
\mathscr {A}_{\alpha }(G,\mathscr {S})
is a finite dimensional semisimple associative algebra associated with
G
,
S
G,\mathscr {S}
, and a
2
2
-cocycle
α
\alpha
naturally determined by the
G
G
-action on
S
\mathscr {S}
. It follows as a natural consequence of the result that for any
g
∈
G
g\in G
every irreducible
g
g
-twisted weak
V
V
-module is a completely reducible weak
V
G
V^G
-module.