You do not amend the W-2 from your employer, you prepare a tax return and make payment.
The Ohio Department of Taxation allows you to prepare a taxing school district tax return at this website.
This Ohio Department of Taxation website explains that you should submit an IT-4 to your employer to have school district taxes withheld from your pay.
“Any individual or estate that receives income while a resident of a taxing school district is subject to school district income tax…..To have school district income tax withheld from your pay, submit an IT-4 with the school district name and number to your employer. Please note, your school district withholding is not the same as your city income tax withholding.”
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