No, only the custodial parent can claim head of household. You are correct. If the nights the child spend with both parents are the same, the tiebreaker is the higher AGI. You would be the custodial parent.
Of note assuming the child is age 16 or under and qualifies for the stimulus payments, the child's mother should also get those payments on her 2020 tax return since she is claiming the child this year. That is true even though you may have received stimulus payments for the child based on the 2019 returns.
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