def METHOD_NAME(self, x, check=True):
    r"""
    Create an element of this Galois group from ``x``.
    INPUT:
    - ``x`` -- one of the following (`G` is this Galois group):
      - the integer 1, denoting the identity of `G`;
      - an element of `G`;
      - a list of length 1, giving the exponent of Frobenius
      - a permutation of the right length that defines an element of `G`,
        or anything that coerces into such a permutation;
      - an automorphism of the finite field.
    - ``check`` -- check that automorphisms have the correct domain and codomain
    EXAMPLES::
        sage: k = GF(3^3)
        sage: G = k.galois_group()
        sage: G(1)
        1
        sage: G([2])
        Frob^2
        sage: G(G.gens()[0])
        Frob
        sage: G([(1,3,2)])
        Frob^2
        sage: G(k.hom(k.gen()^3, k))
        Frob
        sage: G(k.frobenius_endomorphism())
        Frob
    """
    if x == 1:
        return self.element_class(self, [0])
    k = self._field
    d = k.degree()
    n = None
    if isinstance(x, GaloisGroup_GFElement) and x.parent() is self:
        n = x.exponents()[0]
    elif isinstance(x, FiniteFieldHomomorphism_generic):
        if check and not (x.domain() is k and x.codomain() is k):
            raise ValueError("Not an automorphism of the correct finite field")
        a = k.gen()
        b = x(a)
        q = k.base_ring().cardinality()
        n = 0
        while n < d:
            if a == b:
                break
            n += 1
            a = a**q
        else:
            raise RuntimeError("Automorphism was not a power of Frobenius")
    elif isinstance(x, FrobeniusEndomorphism_finite_field):
        if check and not x.domain() is k:
            raise ValueError("Not an automorphism of the correct finite field")
        n = x.power()
    elif isinstance(x, list) and len(x) == 1 and x[0] in ZZ:
        n = x[0]
    else:
        g = self.permutation_group()(x)
        n = g(1) - 1
    return self.element_class(self, [n])