The one substantial ancient report about the book is preserved by Eusebius in the Ecclesiastical History, at book six chapter twelve. Serapion, bishop of Antioch around 199 to 211, visited the congregation at Rhossus in Cilicia, found a gospel circulating under Peter's name, and, without reading it closely, allowed its use. Afterwards he was told it was being used to support heterodox teaching. He obtained a copy from those he calls the successors of Marcianus, read it, and wrote a tract listing the passages he objected to. His conclusion, as Eusebius quotes it, is measured: most of it accords with the true teaching of the Saviour, but some things have been added. The letter uses the term Docetae for the group he associates with the book, which is why the charge is usually summarised as docetism.
Whether the book Serapion read is the book found at Akhmim cannot be established. The Akhmim narrative's suspect phrases are mild, and Serapion says he objected to specific additions he does not fully specify. The identification is probable and is generally assumed, but it remains an inference.
The other part of its afterlife is harder. The text removes Pilate from the execution entirely, has the Jewish authorities refuse the hand-washing that Pilate performs, and has the people of Jerusalem, on seeing the signs at the death, lament their sins and speak of the coming judgement on the city. That is a sharper and more sustained transfer of guilt than anything in the canonical accounts, and it is regularly cited in studies of how Christian passion narratives hardened into an instrument of anti-Jewish polemic. It is worth noting that this, rather than the walking cross, is what made the text theologically consequential, and that the church's rejection of it had nothing to do with that feature.