aut.py

import itertools as it

S4 = list(it.permutations(range(4)))
X = {(0, 1, 2, 3), (1, 2, 0, 3), (2, 0, 1, 3)}

S4_inv = { s: inv for s in S4 for inv in S4 if all([s[inv[n]] == n for n in range(4)]) }

def aut(s, x):
    inv = S4_inv[s]
    return tuple(inv[x[s[n]]] for n in range(4))

Gx = { tuple(sorted(aut(s, x) for x in X)) for s in S4 }
G_x = { s for s in S4 if all(aut(s, x) in X for x in X) }

print(f"Gx: {Gx}")
print(f"G_x: {G_x}")