the 2 isometry classes of irreducible [5,2,3]_4 codes are:

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

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