the 4 isometry classes of irreducible [6,2,5]_16 codes are:

code no       1:
================
1 1 1 1 1 0
4 3 2 1 0 1
the automorphism group has order 2
and is strongly generated by the following 1 elements:
(
0 3 0 0 
10 0 0 0 
12 5 6 3 
6 6 6 6 
, 0
)
acting on the columns of the generator matrix as follows (in order):
(1, 2)(3, 6)(4, 5)
orbits: { 1, 2 }, { 3, 6 }, { 4, 5 }

code no       2:
================
1 1 1 1 1 0
12 3 2 1 0 1
the automorphism group has order 8
and is strongly generated by the following 3 elements:
(
7 0 0 0 
0 7 0 0 
7 7 7 7 
0 0 0 7 
, 2
, 
0 6 0 0 
14 0 0 0 
0 0 3 0 
11 3 6 5 
, 2
, 
3 14 2 12 
0 0 0 2 
0 0 12 0 
13 0 0 0 
, 3
)
acting on the columns of the generator matrix as follows (in order):
(3, 5), 
(1, 2)(4, 6), 
(1, 4, 2, 6)
orbits: { 1, 2, 6, 4 }, { 3, 5 }

code no       3:
================
1 1 1 1 1 0
6 5 2 1 0 1
the automorphism group has order 20
and is strongly generated by the following 2 elements:
(
13 0 0 0 
0 0 0 2 
0 15 0 0 
5 7 8 10 
, 1
, 
0 5 0 0 
0 0 5 0 
0 0 0 5 
5 0 0 0 
, 1
)
acting on the columns of the generator matrix as follows (in order):
(2, 3, 6, 4), 
(1, 4, 3, 2)
orbits: { 1, 2, 4, 3, 6 }, { 5 }

code no       4:
================
1 1 1 1 1 0
10 5 2 1 0 1
the automorphism group has order 12
and is strongly generated by the following 3 elements:
(
2 0 0 0 
4 6 13 7 
0 0 0 1 
0 0 11 0 
, 2
, 
0 4 0 0 
4 0 0 0 
4 4 4 4 
0 0 0 4 
, 2
, 
0 0 2 0 
0 0 0 3 
7 8 2 3 
7 0 0 0 
, 2
)
acting on the columns of the generator matrix as follows (in order):
(2, 6)(3, 4), 
(1, 2)(3, 5), 
(1, 4, 2, 5, 6, 3)
orbits: { 1, 2, 3, 6, 4, 5 }