Is your employer willing to correct it and make the payment to the correct state and issue you a corrected W-2? If so, that would be the easiest way.
If not, you will simply file a nonresident tax return for the state that they paid the taxes to and claim $0 income for that state. This will allow you to claim a refund for that state. You will then need to file a return for the correct state. When you file the return for the correct state, you will then likely need to pay taxes to that state since 6 months worth of taxes were not paid to them.
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