By application of Grimm's law, this hypothetical root has the form *pug–.
Since uncontaminated resources were scarce, supposedly many towns required permission to have children.
Hence, the legend goes, that couples that were having children were required to first obtain royal permission (usually from a local magistrate or lord) and then place a sign somewhere visible from the road in their home that said "Fornicating Under Consent of King," which was later shortened to "FUCK." This story is hard to document, but has persisted in oral and literary traditions for many years; however, it has been demonstrated to be an urban legend.
None of these acronyms was ever recorded before the 1960s, according to the authoritative lexicographical work The F-Word, and thus are backronyms.
In any event, the word fuck has been in use far too long for some of these supposed origins to be possible.
A possible intermediate might be a Latin 4th-declension verbal noun *fūtus, with possible meanings including "act of (pro)creating".