the 4 isometry classes of irreducible [10,5,5]_4 codes are:

code no       1:
================
1 1 1 1 1 1 0 0 0 0
2 2 1 1 0 0 1 0 0 0
3 2 1 0 1 0 0 1 0 0
2 0 2 1 1 0 0 0 1 0
1 3 3 2 1 0 0 0 0 1
the automorphism group has order 16
and is strongly generated by the following 3 elements:
(
3 0 0 0 0 
1 1 1 1 1 
1 0 1 2 2 
2 2 3 3 0 
1 2 3 0 3 
, 1
, 
0 0 0 0 2 
2 2 1 1 0 
0 0 1 0 0 
3 3 3 3 3 
1 3 3 2 1 
, 0
, 
0 0 0 3 0 
0 0 0 0 2 
0 0 2 0 0 
2 3 1 0 1 
3 3 3 3 3 
, 1
)
acting on the columns of the generator matrix as follows (in order):
(2, 6)(3, 9)(4, 7)(5, 8), 
(1, 8, 10, 5)(2, 6, 4, 7), 
(1, 6, 5, 2, 10, 7, 8, 4)
orbits: { 1, 5, 4, 8, 10, 6, 7, 2 }, { 3, 9 }

code no       2:
================
1 1 1 1 1 1 0 0 0 0
2 2 1 1 0 0 1 0 0 0
3 2 1 0 1 0 0 1 0 0
2 0 2 1 1 0 0 0 1 0
3 1 0 3 1 0 0 0 0 1
the automorphism group has order 10
and is strongly generated by the following 2 elements:
(
3 0 0 0 0 
1 1 1 1 1 
1 0 1 2 2 
2 2 3 3 0 
1 2 3 0 3 
, 1
, 
2 2 3 3 0 
0 0 0 3 0 
0 0 1 0 0 
0 3 0 0 0 
3 0 3 1 1 
, 1
)
acting on the columns of the generator matrix as follows (in order):
(2, 6)(3, 9)(4, 7)(5, 8), 
(1, 7)(2, 4)(5, 9)(8, 10)
orbits: { 1, 7, 4, 2, 6 }, { 3, 9, 5, 8, 10 }

code no       3:
================
1 1 1 1 1 1 0 0 0 0
2 2 1 1 0 0 1 0 0 0
3 2 1 0 1 0 0 1 0 0
2 0 3 1 1 0 0 0 1 0
1 2 0 2 1 0 0 0 0 1
the automorphism group has order 10
and is strongly generated by the following 1 elements:
(
3 3 1 1 0 
1 2 3 0 3 
0 0 0 0 2 
0 0 3 0 0 
1 1 1 1 1 
, 1
)
acting on the columns of the generator matrix as follows (in order):
(1, 6, 5, 3, 4, 9, 8, 2, 10, 7)
orbits: { 1, 7, 10, 2, 8, 9, 4, 3, 5, 6 }

code no       4:
================
1 1 1 1 1 1 0 0 0 0
2 2 1 1 0 0 1 0 0 0
3 2 1 0 1 0 0 1 0 0
2 0 3 1 1 0 0 0 1 0
1 2 3 2 1 0 0 0 0 1
the automorphism group has order 8
and is strongly generated by the following 2 elements:
(
0 3 0 0 0 
0 0 3 0 0 
3 2 1 0 1 
1 1 1 1 1 
0 0 0 0 1 
, 0
, 
2 2 2 2 2 
2 0 1 3 3 
2 1 3 1 2 
0 0 1 0 0 
0 0 0 0 2 
, 1
)
acting on the columns of the generator matrix as follows (in order):
(1, 8, 3, 2)(4, 10, 9, 6), 
(1, 9, 2, 10, 3, 4, 8, 6)
orbits: { 1, 2, 6, 3, 9, 8, 10, 4 }, { 5 }, { 7 }