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This is my attempt:

    R = PolynomialRing(QQ, 4, ["q1","q2","q3","q4"])
    q1,q2,q3,q4 = R.gens()
    f = 1+q1+q2 # for example
    G = SymmetricGroup(4)
    cl = G.conjugacy_classes_subgroups()
    r = []
    for g in G:
        if (f * g) == f: r.append(g)
    H = PermutationGroup(r)
    for p in cl:
        if H.is_isomorphic(p): return p.id()

This is my attempt:attempt (edit):

 R = PolynomialRing(QQ, 4, ["q1","q2","q3","q4"])
 q1,q2,q3,q4 = R.gens()
 f = 1+q1+q2 # for example
 G = SymmetricGroup(4)
    cl = G.conjugacy_classes_subgroups()
    r = []
 for g in G:
     if (f * g) == f: r.append(g)
 H = PermutationGroup(r)
    for p in cl:
        if H.is_isomorphic(p): return p.id()
ConjugacyClass(H, G)