the 2 isometry classes of irreducible [6,2,3]_5 codes are:

code no       1:
================
1 1 1 1 4 0
1 1 0 0 0 4
the automorphism group has order 12
and is strongly generated by the following 3 elements:
(
1 0 0 0 
0 1 0 0 
0 0 1 0 
4 4 4 4 
, 
1 0 0 0 
0 1 0 0 
0 0 0 1 
0 0 1 0 
, 
0 1 0 0 
1 0 0 0 
0 0 1 0 
0 0 0 1 
)
acting on the columns of the generator matrix as follows (in order):
(4, 5), 
(3, 4), 
(1, 2)
orbits: { 1, 2 }, { 3, 4, 5 }, { 6 }

code no       2:
================
1 1 1 1 4 0
2 1 0 0 0 4
the automorphism group has order 12
and is strongly generated by the following 3 elements:
(
3 0 0 0 
0 3 0 0 
0 0 3 0 
2 2 2 2 
, 
4 0 0 0 
0 4 0 0 
1 1 1 1 
0 0 0 4 
, 
2 0 0 0 
1 3 0 0 
2 2 2 2 
0 0 3 0 
)
acting on the columns of the generator matrix as follows (in order):
(4, 5), 
(3, 5), 
(2, 6)(3, 4, 5)
orbits: { 1 }, { 2, 6 }, { 3, 5, 4 }