Usually only FUTA is paid during the year for a household employee. All taxes are then paid using the sch H for the employee and employer. If you paid more than $1,000 in any quarter, you do not file sch H.
See:
2021 Schedule H (Form 1040) - Internal Revenue Service.
2021 Instructions for Schedule H - IRS
and verify that this form is required. Also, verify that you paid Medicare and Social Security tax through the payroll provider rather than the standard Unemployment.
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