the 4 isometry classes of irreducible [10,5,4]_2 codes are:

code no       1:
================
1 1 1 1 1 1 0 0 0 0
1 1 1 0 0 0 1 0 0 0
1 1 0 1 0 0 0 1 0 0
1 0 1 1 0 0 0 0 1 0
0 1 1 1 0 0 0 0 0 1
the automorphism group has order 384
and is strongly generated by the following 5 elements:
(
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
0 0 0 1 0 
1 1 1 1 1 
, 
1 0 0 0 0 
0 1 0 0 0 
0 0 0 1 0 
0 0 1 0 0 
0 0 0 0 1 
, 
1 0 0 0 0 
0 1 0 0 0 
1 1 1 0 0 
1 1 0 1 0 
0 0 0 0 1 
, 
1 0 0 0 0 
1 1 0 1 0 
1 0 1 1 0 
0 0 0 1 0 
0 0 0 0 1 
, 
1 1 1 0 0 
0 1 1 1 0 
0 0 1 0 0 
0 1 0 0 0 
0 0 0 0 1 
)
acting on the columns of the generator matrix as follows (in order):
(5, 6), 
(3, 4)(7, 8), 
(3, 7)(4, 8), 
(2, 8)(3, 9), 
(1, 9, 7)(2, 4, 10)
orbits: { 1, 7, 8, 3, 9, 4, 2, 10 }, { 5, 6 }

code no       2:
================
1 1 1 1 1 1 0 0 0 0
1 1 1 0 0 0 1 0 0 0
1 1 0 1 0 0 0 1 0 0
1 0 1 1 0 0 0 0 1 0
1 1 0 0 1 0 0 0 0 1
the automorphism group has order 48
and is strongly generated by the following 5 elements:
(
1 0 0 0 0 
0 1 0 0 0 
0 0 0 1 0 
0 0 1 0 0 
0 0 0 0 1 
, 
1 0 0 0 0 
0 1 0 0 0 
1 1 1 0 0 
1 1 0 1 0 
0 0 0 0 1 
, 
1 0 0 0 0 
1 0 1 1 0 
0 0 1 0 0 
1 1 1 0 0 
1 1 1 1 1 
, 
1 0 1 1 0 
1 0 0 0 0 
0 0 0 1 0 
1 1 1 0 0 
1 1 0 0 1 
, 
1 1 1 1 1 
1 1 0 0 1 
1 1 0 1 0 
0 0 1 0 0 
1 0 0 0 0 
)
acting on the columns of the generator matrix as follows (in order):
(3, 4)(7, 8), 
(3, 7)(4, 8), 
(2, 9)(4, 7)(5, 6), 
(1, 2, 9)(3, 7, 4)(5, 6, 10), 
(1, 5, 9, 10, 2, 6)(3, 4, 8)
orbits: { 1, 9, 6, 2, 5, 10 }, { 3, 4, 7, 8 }

code no       3:
================
1 1 1 1 1 1 0 0 0 0
1 1 1 0 0 0 1 0 0 0
1 1 0 1 0 0 0 1 0 0
1 0 1 0 1 0 0 0 1 0
1 0 0 1 1 0 0 0 0 1
the automorphism group has order 720
and is strongly generated by the following 8 elements:
(
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
1 1 0 1 0 
1 0 1 0 1 
, 
1 0 0 0 0 
0 1 0 0 0 
1 1 0 1 0 
1 1 1 0 0 
1 1 1 1 1 
, 
1 0 0 0 0 
0 1 0 0 0 
0 0 0 1 0 
0 0 1 0 0 
0 0 0 0 1 
, 
1 0 0 0 0 
0 0 1 0 0 
0 1 0 0 0 
0 0 0 0 1 
0 0 0 1 0 
, 
1 0 0 0 0 
1 1 1 1 1 
1 0 0 1 1 
1 1 0 1 0 
0 0 0 1 0 
, 
0 0 0 0 1 
1 1 1 1 1 
0 0 1 0 0 
0 0 0 1 0 
1 0 0 0 0 
, 
1 0 1 0 1 
1 0 0 0 0 
0 0 0 0 1 
1 1 0 1 0 
1 1 1 0 0 
, 
1 0 0 1 1 
0 0 1 0 0 
1 0 1 0 1 
1 1 1 1 1 
1 1 1 0 0 
)
acting on the columns of the generator matrix as follows (in order):
(4, 8)(5, 9)(6, 10), 
(3, 8)(4, 7)(5, 6), 
(3, 4)(7, 8)(9, 10), 
(2, 3)(4, 5)(8, 9), 
(2, 10, 3, 6)(4, 5, 9, 8), 
(1, 5)(2, 6)(7, 8), 
(1, 2, 9)(3, 7, 5)(4, 6, 8), 
(1, 10)(2, 8, 9, 3)(4, 7, 5, 6)
orbits: { 1, 5, 9, 10, 6, 4, 7, 8, 2, 3 }

code no       4:
================
1 1 1 0 0 1 0 0 0 0
1 1 0 1 0 0 1 0 0 0
1 1 0 0 1 0 0 1 0 0
1 0 1 1 1 0 0 0 1 0
0 1 1 1 1 0 0 0 0 1
the automorphism group has order 1920
and is strongly generated by the following 9 elements:
(
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
0 0 0 1 0 
1 0 1 1 1 
, 
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
0 0 0 1 0 
1 1 0 0 1 
, 
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
1 1 0 1 0 
1 1 0 0 1 
, 
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
1 1 0 0 1 
1 1 0 1 0 
, 
1 0 0 0 0 
0 1 0 0 0 
0 0 1 0 0 
0 1 1 1 1 
0 0 0 1 0 
, 
1 0 0 0 0 
0 1 0 0 0 
1 1 1 0 0 
1 1 0 1 0 
0 0 0 0 1 
, 
1 0 0 0 0 
0 1 0 0 0 
0 0 0 1 0 
0 0 1 0 0 
0 0 0 0 1 
, 
0 0 0 0 1 
1 1 0 0 1 
0 1 0 0 0 
0 0 0 1 0 
1 1 1 0 0 
, 
0 1 1 1 1 
1 0 1 1 1 
0 0 0 1 0 
0 0 1 0 0 
1 1 0 0 1 
)
acting on the columns of the generator matrix as follows (in order):
(5, 9)(8, 10), 
(5, 8)(9, 10), 
(4, 7)(5, 8), 
(4, 8)(5, 7), 
(4, 5, 10)(7, 8, 9), 
(3, 6)(4, 7), 
(3, 4)(6, 7), 
(1, 6, 5)(2, 3, 8), 
(1, 9, 2, 10)(3, 4)(5, 8)(6, 7)
orbits: { 1, 5, 10, 9, 8, 7, 4, 6, 2, 3 }