the 4 isometry classes of irreducible [9,4,4]_2 codes are:

code no       1:
================
1 1 1 1 1 1 0 0 0
1 1 1 0 0 0 1 0 0
1 1 0 1 0 0 0 1 0
1 0 1 1 0 0 0 0 1
the automorphism group has order 48
and is strongly generated by the following 5 elements:
(
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
0 0 0 1 0 
1 1 1 1 1 
, 
1 0 0 0 0 
0 1 0 0 0 
0 0 0 1 0 
0 0 1 0 0 
0 0 0 0 1 
, 
1 0 0 0 0 
0 1 0 0 0 
1 1 1 0 0 
1 1 0 1 0 
0 0 0 0 1 
, 
1 0 0 0 0 
1 0 1 1 0 
0 0 0 1 0 
1 1 0 1 0 
0 0 0 0 1 
, 
1 0 0 0 0 
1 1 0 1 0 
1 0 1 1 0 
0 0 0 1 0 
0 0 0 0 1 
)
acting on the columns of the generator matrix as follows (in order):
(5, 6), 
(3, 4)(7, 8), 
(3, 7)(4, 8), 
(2, 9)(3, 7, 8, 4), 
(2, 8)(3, 9)
orbits: { 1 }, { 2, 9, 8, 3, 7, 4 }, { 5, 6 }

code no       2:
================
1 1 1 1 1 1 0 0 0
1 1 1 0 0 0 1 0 0
1 1 0 1 0 0 0 1 0
1 0 1 0 1 0 0 0 1
the automorphism group has order 72
and is strongly generated by the following 5 elements:
(
1 0 0 0 0 
0 1 0 0 0 
1 1 0 1 0 
1 1 1 0 0 
1 1 1 1 1 
, 
1 0 0 0 0 
0 0 1 0 0 
0 1 0 0 0 
0 0 0 0 1 
0 0 0 1 0 
, 
0 0 0 0 1 
1 1 1 1 1 
0 0 1 0 0 
0 0 0 1 0 
1 0 0 0 0 
, 
1 1 1 1 1 
0 0 0 0 1 
0 0 0 1 0 
0 0 1 0 0 
0 1 0 0 0 
, 
1 0 1 0 1 
1 0 0 0 0 
0 0 0 0 1 
1 1 0 1 0 
1 1 1 0 0 
)
acting on the columns of the generator matrix as follows (in order):
(3, 8)(4, 7)(5, 6), 
(2, 3)(4, 5)(8, 9), 
(1, 5)(2, 6)(7, 8), 
(1, 6)(2, 5)(3, 4), 
(1, 2, 9)(3, 7, 5)(4, 6, 8)
orbits: { 1, 5, 6, 9, 4, 2, 7, 8, 3 }

code no       3:
================
1 1 1 0 0 1 0 0 0
1 1 0 1 0 0 1 0 0
1 0 1 1 0 0 0 1 0
0 1 1 1 1 0 0 0 1
the automorphism group has order 336
and is strongly generated by the following 6 elements:
(
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
0 0 0 1 0 
0 1 1 1 1 
, 
1 0 0 0 0 
0 1 0 0 0 
1 1 1 0 0 
1 1 0 1 0 
0 0 0 0 1 
, 
1 0 0 0 0 
0 1 0 0 0 
1 1 0 1 0 
1 1 1 0 0 
0 0 0 0 1 
, 
1 0 0 0 0 
1 1 0 1 0 
1 1 1 0 0 
0 1 0 0 0 
0 0 0 0 1 
, 
1 1 1 0 0 
0 0 1 0 0 
0 1 0 0 0 
0 0 0 1 0 
0 0 0 0 1 
, 
1 1 0 1 0 
0 0 0 1 0 
1 1 1 0 0 
1 0 0 0 0 
0 0 0 0 1 
)
acting on the columns of the generator matrix as follows (in order):
(5, 9), 
(3, 6)(4, 7), 
(3, 7)(4, 6), 
(2, 4, 7)(3, 8, 6), 
(1, 6)(2, 3), 
(1, 4, 2, 7)(3, 6)
orbits: { 1, 6, 7, 3, 4, 8, 2 }, { 5, 9 }

code no       4:
================
1 1 1 0 0 1 0 0 0
1 1 0 1 0 0 1 0 0
1 1 0 0 1 0 0 1 0
1 0 1 1 1 0 0 0 1
the automorphism group has order 192
and is strongly generated by the following 8 elements:
(
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
1 1 0 1 0 
1 1 0 0 1 
, 
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
0 0 0 0 1 
0 0 0 1 0 
, 
1 0 0 0 0 
0 1 0 0 0 
0 0 0 1 0 
0 0 1 0 0 
0 0 0 0 1 
, 
1 0 0 0 0 
0 1 0 0 0 
1 1 1 0 0 
1 1 0 1 0 
0 0 0 0 1 
, 
1 0 0 0 0 
0 1 0 0 0 
1 1 0 0 1 
0 0 1 0 0 
1 1 0 1 0 
, 
0 1 0 0 0 
1 0 0 0 0 
0 0 1 0 0 
1 1 0 1 0 
0 0 0 0 1 
, 
1 1 1 0 0 
0 0 1 0 0 
1 0 0 0 0 
1 1 0 1 0 
0 0 0 0 1 
, 
0 0 0 0 1 
1 1 0 0 1 
0 1 0 0 0 
0 0 0 1 0 
1 1 1 0 0 
)
acting on the columns of the generator matrix as follows (in order):
(4, 7)(5, 8), 
(4, 5)(7, 8), 
(3, 4)(6, 7), 
(3, 6)(4, 7), 
(3, 4, 8)(5, 6, 7), 
(1, 2)(4, 7), 
(1, 3, 2, 6)(4, 7), 
(1, 6, 5)(2, 3, 8)
orbits: { 1, 2, 6, 5, 3, 8, 7, 4 }, { 9 }