They are a 19-year-old, a 46-year-old and a 49-year-old, all from South Norwood, a 49-year-old from Lambeth and a 55-year-old from Beckenham.
It showed a large flammable model marked "Grenfell Tower", complete with paper figures of people at their windows, before it was set on fire.
Khadijah Mamudu, whose mother and younger brother escaped the fire, told the Press Association: "Their actions upset so many people around the world and they need to think long and hard".
The video also drew allegations that it was racist because numerous Grenfell Tower victims were people of immigrant backgrounds and Muslim. After it's set alight, the group can be heard chuckling and cracking jokes.
The video is believed to have been circulated originally in closed WhatsApp groups before being posted to social media and shared widely on Twitter and Facebook, at which point it was reported to the police.
Another says: "Did it start on the 10th floor?"
'Elhouhabi shamelessly took advantage of the efforts to rehouse people whose homes had been destroyed by claiming his link to one of the flats, and he was subsequently put in hotel accommodation.
One person can be hear saying: "That little ninja is really getting it", which people understand to be referring to a cardboard Muslim woman wearing a hijab.
Earlier, Scotland Yard has confirmed that five men presented themselves to a police station after an investigation into the video has been launched.
Prime Minister Theresa May tweeted: "To disrespect those who lost their lives at Grenfell Tower, as well as their families and loved ones, is utterly unacceptable".
The Leader of the Royal Borough of Kensington and Chelsea Council, where Grenfell Tower is located, called the video "absolutely disgusting" in a statement.
"It can not be justified on any level".
A video has emerged on social media showing a group laughing at the victims of the fire tragedy while burning a cardboard model.
MEND, an anti-Islamophobia organization, said the "sickening Islamophobic" video amounted to a hate crime.
And the Justice4Grenfell movement said it was "disgusted and shocked at the inhumanity and callousness of those involved in this video".