the 3 isometry classes of irreducible [16,7,6]_2 codes are:

code no       1:
================
1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0
1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0
1 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0
1 0 1 0 1 0 1 1 0 0 0 0 0 1 0 0
1 0 1 1 0 1 0 0 1 0 0 0 0 0 1 0
1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 1
the automorphism group has order 16
and is strongly generated by the following 3 elements:
(
1 0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 1 0 
1 0 1 0 1 0 1 1 0 
1 1 0 1 0 1 0 1 0 
1 1 1 1 1 1 1 1 1 
0 0 0 0 0 1 0 0 0 
1 0 1 1 0 1 0 0 1 
0 1 0 0 0 0 0 0 0 
1 1 1 1 1 0 0 0 0 
, 
0 0 0 0 0 0 0 0 1 
0 1 0 0 0 0 0 0 0 
1 0 1 1 0 1 0 0 1 
0 0 0 0 0 1 0 0 0 
1 1 1 1 1 0 0 0 0 
0 0 0 1 0 0 0 0 0 
1 1 1 0 0 1 1 0 0 
1 0 1 0 1 0 1 1 0 
1 0 0 0 0 0 0 0 0 
, 
1 1 1 1 1 1 1 1 1 
0 0 0 0 0 0 1 0 0 
0 0 0 0 0 0 0 1 0 
0 0 0 0 0 0 0 0 1 
0 0 0 0 0 1 0 0 0 
0 0 0 0 1 0 0 0 0 
0 1 0 0 0 0 0 0 0 
0 0 1 0 0 0 0 0 0 
0 0 0 1 0 0 0 0 0 
)
acting on the columns of the generator matrix as follows (in order):
(2, 8)(3, 14)(4, 13)(5, 10)(7, 15)(9, 11)(12, 16), 
(1, 9)(3, 15)(4, 6)(5, 11)(7, 12)(8, 14)(10, 13), 
(1, 10)(2, 7)(3, 8)(4, 9)(5, 6)(12, 15)(14, 16)
orbits: { 1, 9, 10, 11, 4, 5, 13, 6 }, { 2, 8, 7, 14, 3, 15, 12, 16 }

code no       2:
================
1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0
1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0
1 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0
1 0 1 0 1 0 1 1 0 0 0 0 0 1 0 0
0 1 1 0 1 1 0 0 1 0 0 0 0 0 1 0
1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 1
the automorphism group has order 48
and is strongly generated by the following 4 elements:
(
1 0 0 0 0 0 0 0 0 
0 1 0 0 0 0 0 0 0 
1 1 0 0 1 0 0 1 1 
0 0 0 1 0 0 0 0 0 
1 1 1 0 0 1 1 0 0 
0 0 0 0 0 0 0 1 0 
0 0 0 0 0 0 0 0 1 
0 0 0 0 0 1 0 0 0 
0 0 0 0 0 0 1 0 0 
, 
1 0 0 0 0 0 0 0 0 
1 0 1 0 1 0 1 1 0 
0 0 0 0 0 0 1 0 0 
1 1 0 1 0 1 0 1 0 
0 0 0 0 0 1 0 0 0 
0 0 0 0 1 0 0 0 0 
0 0 1 0 0 0 0 0 0 
1 1 1 0 0 1 1 0 0 
1 1 0 0 1 0 0 1 1 
, 
1 0 1 0 1 0 1 1 0 
0 1 1 0 1 1 0 0 1 
0 0 1 0 0 0 0 0 0 
0 0 0 1 0 0 0 0 0 
0 0 0 0 1 0 0 0 0 
0 0 0 0 0 0 0 0 1 
0 0 0 0 0 0 0 1 0 
0 0 0 0 0 0 1 0 0 
0 0 0 0 0 1 0 0 0 
, 
0 0 0 0 0 0 1 0 0 
0 0 0 0 0 0 0 0 1 
0 0 0 0 1 0 0 0 0 
1 1 0 1 0 1 0 1 0 
0 0 1 0 0 0 0 0 0 
0 1 1 0 1 1 0 0 1 
1 0 0 0 0 0 0 0 0 
1 0 1 0 1 0 1 1 0 
0 1 0 0 0 0 0 0 0 
)
acting on the columns of the generator matrix as follows (in order):
(3, 16)(5, 12)(6, 8)(7, 9)(10, 11), 
(2, 14)(3, 7)(4, 13)(5, 6)(8, 12)(9, 16), 
(1, 14)(2, 15)(6, 9)(7, 8)(10, 11), 
(1, 7)(2, 9)(3, 5)(4, 13)(6, 15)(8, 14)(10, 11)
orbits: { 1, 14, 7, 2, 8, 9, 3, 15, 6, 12, 16, 5 }, { 4, 13 }, { 10, 11 }

code no       3:
================
1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0
1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0
1 0 0 1 1 1 0 1 0 0 0 0 1 0 0 0
0 1 0 1 0 1 1 1 0 0 0 0 0 1 0 0
0 1 0 1 1 0 1 0 1 0 0 0 0 0 1 0
1 0 0 0 1 1 1 0 1 0 0 0 0 0 0 1
the automorphism group has order 5760
and is strongly generated by the following 7 elements:
(
1 0 0 0 0 0 0 0 0 
0 1 0 0 0 0 0 0 0 
0 0 1 0 0 0 0 0 0 
0 0 0 0 0 0 1 0 0 
0 0 0 0 0 1 0 0 0 
0 0 0 0 1 0 0 0 0 
0 0 0 1 0 0 0 0 0 
0 0 0 0 0 0 0 0 1 
0 0 0 0 0 0 0 1 0 
, 
1 0 0 0 0 0 0 0 0 
0 1 0 0 0 0 0 0 0 
1 0 0 1 1 1 0 1 0 
0 1 0 1 0 1 1 1 0 
0 0 0 0 1 0 0 0 0 
1 1 1 1 1 1 1 1 1 
1 1 1 1 1 0 0 0 0 
0 0 0 0 0 0 0 0 1 
0 0 0 0 0 0 0 1 0 
, 
1 0 0 0 0 0 0 0 0 
0 1 0 0 0 0 0 0 0 
0 0 0 0 0 0 1 0 0 
1 1 1 0 0 1 1 0 0 
0 0 1 0 0 0 0 0 0 
0 1 0 1 0 1 1 1 0 
0 0 0 0 1 0 0 0 0 
0 0 0 0 0 0 0 1 0 
0 0 0 0 0 0 0 0 1 
, 
1 0 0 0 0 0 0 0 0 
0 0 0 0 0 0 1 0 0 
1 0 0 1 1 1 0 1 0 
0 0 0 0 0 0 0 1 0 
1 1 1 0 0 1 1 0 0 
0 1 0 1 0 1 1 1 0 
0 1 0 0 0 0 0 0 0 
0 0 0 1 0 0 0 0 0 
1 1 1 1 1 1 1 1 1 
, 
1 0 0 0 0 0 0 0 0 
0 0 0 0 1 0 0 0 0 
0 0 1 0 0 0 0 0 0 
1 1 1 1 1 0 0 0 0 
0 1 0 0 0 0 0 0 0 
1 1 1 1 1 1 1 1 1 
0 1 0 1 0 1 1 1 0 
0 1 0 1 1 0 1 0 1 
1 1 1 0 0 1 1 0 0 
, 
1 1 1 0 0 1 1 0 0 
1 1 1 1 1 0 0 0 0 
0 0 0 0 0 0 0 1 0 
0 1 0 1 0 1 1 1 0 
0 0 0 0 1 0 0 0 0 
1 0 0 1 1 1 0 1 0 
0 0 0 0 0 0 1 0 0 
0 0 1 0 0 0 0 0 0 
1 1 1 1 1 1 1 1 1 
, 
1 0 0 0 1 1 1 0 1 
1 1 1 1 1 0 0 0 0 
1 0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 1 
0 0 0 1 0 0 0 0 0 
0 0 0 0 0 0 0 1 0 
1 1 1 1 1 1 1 1 1 
0 0 0 0 0 0 1 0 0 
0 1 0 0 0 0 0 0 0 
)
acting on the columns of the generator matrix as follows (in order):
(4, 7)(5, 6)(8, 9)(11, 12)(13, 16)(14, 15), 
(3, 13)(4, 14)(6, 10)(7, 11)(8, 9)(12, 15), 
(3, 5, 7)(4, 13, 12)(6, 11, 14)(10, 16, 15), 
(2, 7)(3, 13)(4, 8)(5, 12)(6, 14)(9, 10), 
(2, 5)(4, 11)(6, 10)(7, 14)(8, 15)(9, 12), 
(1, 12)(2, 11)(3, 8)(4, 14)(6, 13)(9, 10), 
(1, 3, 16)(2, 9, 4, 5, 12, 11)(6, 14, 15, 10, 7, 8)
orbits: { 1, 12, 16, 11, 15, 13, 5, 9, 10, 3, 7, 6, 4, 2, 14, 8 }