Falsely

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Old English fals ‘fraud, deceit’, from Latin falsum ‘fraud’, neuter past participle of fallere ‘deceive’; reinforced or re-formed in Middle English from Old French fals, faus ‘false’.

wiktionary

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false +‎  -ly

etymonline

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falsely (adv.)

c. 1200, "with intent to deceive, deceitfully," from false + -ly (2). From c. 1300 as "wrongly; untruthfully;" early 14c. as "incorrectly."