The woman, 23-year-old Korryn Gaines, was later shot dead by the police. On her account, she was recounting the standoff that led to the injury of her son and, ultimately, her death.
The incident occurred on 3 August and police claimed they made their request to Facebook after Gaines' followers urged her not to comply with negotiators' bids to make her surrender peacefully, according to NBC News.
Police began shooting after Gaines fired two rounds at them. None of the police involved were wearing body cameras, according to NBC.
Gaines' death comes after a number of cases where police around the US have shot black people dead. Many of the incidents have led to protests and, in one case, to retaliation when police officers were shot dead.
SumOfUs said that if Facebook chose to comply with police requests to suspend video broadcasts and accounts, it would end up shield police misbehaviour.
The group has started a petition to try and make Facebook change its behaviour.
"If we pile on the pressure, we could force (Facebook chief executive) Mark Zuckerberg to go beyond words to action – and make Facebook change its policy," it said in a media release.
"Our pressure is already working. Over 60,000 SumOfUs members in the United States and Canada have signed our petition, and our campaign is making headlines across the planet."
The group added that in recent years, social media and shareable video had helped create awareness about "the ongoing epidemic of police violence especially against black communities in the United States and ongoing racial tensions in Europe".
"But if Facebook censors this critical tool at the behest of the police, that could all change. That's why we need Facebook to explain its actions and stop this dangerous precedent immediately."