the 3 isometry classes of irreducible [6,2,4]_4 codes are:

code no       1:
================
1 1 1 1 1 0
2 1 1 0 0 1
the automorphism group has order 16
and is strongly generated by the following 4 elements:
(
2 0 0 0 
0 2 0 0 
0 0 2 0 
2 2 2 2 
, 0
, 
1 0 0 0 
1 1 1 1 
0 0 0 1 
0 0 1 0 
, 1
, 
2 0 0 0 
0 0 2 0 
0 2 0 0 
2 2 2 2 
, 0
, 
2 3 3 0 
0 1 0 0 
0 0 1 0 
2 2 2 2 
, 1
)
acting on the columns of the generator matrix as follows (in order):
(4, 5), 
(2, 5)(3, 4), 
(2, 3)(4, 5), 
(1, 6)(4, 5)
orbits: { 1, 6 }, { 2, 5, 3, 4 }

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

code no       3:
================
1 1 1 0 1 0
1 1 0 1 0 1
the automorphism group has order 96
and is strongly generated by the following 7 elements:
(
1 0 0 0 
0 1 0 0 
0 0 1 0 
0 0 0 1 
, 1
, 
1 0 0 0 
0 1 0 0 
0 0 1 0 
1 1 0 1 
, 0
, 
2 0 0 0 
0 2 0 0 
2 2 2 0 
0 0 0 2 
, 1
, 
2 0 0 0 
0 2 0 0 
2 2 0 2 
2 2 2 0 
, 0
, 
0 1 0 0 
1 0 0 0 
0 0 1 0 
0 0 0 1 
, 1
, 
1 1 1 0 
0 0 1 0 
1 0 0 0 
0 0 0 1 
, 0
, 
2 2 0 2 
0 0 0 2 
2 0 0 0 
0 0 2 0 
, 0
)
acting on the columns of the generator matrix as follows (in order):
id, 
(4, 6), 
(3, 5), 
(3, 6)(4, 5), 
(1, 2), 
(1, 3, 2, 5), 
(1, 3, 4, 2, 5, 6)
orbits: { 1, 2, 5, 6, 3, 4 }