the 4 isometry classes of irreducible [6,2,4]_5 codes are:

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

code no       2:
================
1 1 1 1 4 0
3 1 1 0 0 4
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 
3 3 3 3 
, 
4 0 0 0 
0 0 0 4 
1 1 1 1 
0 4 0 0 
, 
3 0 0 0 
2 2 2 2 
0 0 0 3 
0 3 0 0 
, 
1 2 2 0 
0 4 0 0 
0 0 4 0 
0 0 0 1 
)
acting on the columns of the generator matrix as follows (in order):
(4, 5), 
(2, 4)(3, 5), 
(2, 4, 3, 5), 
(1, 6)
orbits: { 1, 6 }, { 2, 4, 5, 3 }

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

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