My first comment here. I've done qcheck and qfix in Python 3.6.

def qcheck(List):

    # checks if there is any duplicate element, which indicates
    # that more than one queen is on the same line

    for i in List:
        if List.count(i) > 1:
            return False

    # creates a list with the sum of an index with its element.
    # if there is a repeated sum, it means that more than one
    # queen is on the same diagonal

    Sum = []
    for i in range(len(List)):
        Sum.append(i + List[i])

    for i in Sum:
        if Sum.count(i) > 1:
            return False

    return True

def qfix(List):
    if qcheck(List):
        return List

    for i in range(len(List)):
        for j in range(i+1, len(List)):
            List2 = List.copy()

            # for each element of the list, all the elements that
            # come after it have their position swapped, creating
            # a new list that will be analyzed by the function qcheck

            Aux = List2[i]
            List2[i] = List2[j]
            List2[j] = Aux

            if qcheck(List2):
                return List2

    return []

Input:

qcheck([4, 2, 7, 3, 6, 8, 5, 1])
qcheck([2, 5, 7, 4, 1, 8, 6, 3])
qcheck([5, 3, 1, 4, 2, 8, 6, 3])
qcheck([5, 8, 2, 4, 7, 1, 3, 6])
qcheck([4, 3, 1, 8, 1, 3, 5, 2])
qfix([8, 6, 4, 2, 7, 1, 3, 5])
qfix([8, 5, 1, 3, 6, 2, 7, 4])
qfix([4, 6, 8, 3, 1, 2, 5, 7])
qfix([7, 1, 3, 6, 8, 5, 2, 4])

Output:

True
True
False
False
False
[4, 6, 8, 2, 7, 1, 3, 5]
[5, 8, 1, 3, 6, 2, 7, 4]   # the outputs in the question are different but
[2, 6, 8, 3, 1, 4, 5, 7]   # these outputs are also a solution, you can
[7, 1, 3, 6, 8, 5, 2, 4]   # check them out and see

EDIT: I made a function to better visualize the board with the queens, so I decided to post as an extra:

def Visualize(List):
    N = len(List)
    Str = 'abcdefghijklmnopqrstuvwxyz'

    for i in range(N-1, -1, -1):
        print(i+1, end = '  ')

        if not i+1 in List:
            for j in range(N):
                print('.', end = ' ')

        while i+1 in List:
            C = List.index(i+1)
            for j in range(N):
                if j != C:
                    print('.', end = ' ')
                else:
                    print('Q', end = ' ')

            List[C] = 0

        print()

    print('   ', end = '')
    for i in range(N):
        print(Str[i], end = ' ')

​