We consider a set
S
P
G
(
A
)
SPG(\mathcal {A})
of pure split states on a quantum spin chain
A
\mathcal {A}
which are invariant under the on-site action
τ
\tau
of a finite group
G
G
. For each element
ω
\omega
in
S
P
G
(
A
)
SPG(\mathcal {A})
we can associate a second cohomology class
c
ω
,
R
c_{\omega ,R}
of
G
G
. We consider a classification of
S
P
G
(
A
)
SPG(\mathcal {A})
whose criterion is given as follows:
ω
0
\omega _{0}
and
ω
1
\omega _{1}
in
S
P
G
(
A
)
SPG(\mathcal {A})
are equivalent if there are automorphisms
Ξ
R
\Xi _{R}
,
Ξ
L
\Xi _L
on
A
R
\mathcal {A}_{R}
,
A
L
\mathcal {A}_{L}
(right and left half infinite chains) preserving the symmetry
τ
\tau
, such that
ω
1
\omega _{1}
and
ω
0
∘
(
Ξ
L
⊗
Ξ
R
)
\omega _{0}\circ \left ( \Xi _{L}\otimes \Xi _{R}\right )
are quasi-equivalent. It means that we can move
ω
0
\omega _{0}
close to
ω
1
\omega _{1}
without changing the entanglement nor breaking the symmetry. We show that the second cohomology class
c
ω
,
R
c_{\omega ,R}
is the complete invariant of this classification.