Edwards returns to Anfield having left as sporting director in 2022.
"Michael is one of the most formidable executive talents in world football," said FSG president Mike Gordon.
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"He returns to us in a role with greater seniority than he held previously and with a wider remit.
"It was clear when we initially reached out to him that a broader scope would be a key motivating factor for any potential return to the industry. In this regard - and many others - his ambitions matched ours."
Gordon also said Edwards "will take over responsibilities I have previously held regarding oversight and management of Liverpool FC's football operation".