Police searching for two-year-old Chayson Basinio knew it was a race against time to find the missing child, who had reportedly disappeared from a supermarket car park.

The local judge opened an inquiry for kidnapping and sequestration and police divers dredged a lake, fearing the child may have drowned.

As the days passed without any leads or clues, detectives at Moulins in the Auvergne prepared to warn relatives who had alerted them that they could find no trace of the boy.

Which, in the circumstances, was hardly surprising. In fact, neither Chayson Basinio nor his parents existed – except in the virtual world of social media.

Police had found photographs allegedly of the boy and his father, Rayane Basinio, 20, on Facebook, but absolutely no evidence that they were real.

Eric Mazaud, the public prosecutor, said the investigation had changed, but charges would be made.

"The inquiry for kidnapping and sequestration has obviously been redirected into one of reporting an imaginary crime or offence," Mazaud said.

"It [the inquiry] was long and complicated but we can now say that the young Chayson has never existed and nor have his father or mother."

The boy was reported missing on Friday by a woman who claimed to be the boy's great-aunt. She told officers she had last seen the child the previous week near a supermarket and believed he had been kidnapped. She told detectives that Basinio and the boy's mother had separated and she had no idea where they were.

Detectives continued to search for the boy, but became suspicious after noting inconsistencies in her story. The woman, who has not been named, is now in police custody. She faces up to six months in prison and a €7,500 (£6,180) fine if found guilty of inventing a crime.

Police said her teenage daughter and a cousin – both minors – who are believed to have set up the false Facebook account and pirated pictures from other accounts on the site were also being questioned.

"Sadly, this is a very modern-day story. Someone decided to create false Facebook accounts and took pictures from real accounts to feed the false accounts and make these people seem real," Mazaud added.

He said the imaginary family had been created several months ago.

"We don't know why someone decided to make a false report. We are currently trying to work out a motive. Either that person has a psychological problem, or there were other purposes behind it, such as revenge."