It is not known if the kitten died, nor who shot the footage - but Paw Justice said on Facebook it was believed to have been filmed in Invercargill.
The video quickly spread online with more than 100,000 views in a matter of hours last night.
Within an hour an Invercargill teen had been named as the girl in the video and posters were threatening violence against her.
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In its Facebook post Paw Justice said the video "sickens and saddens us at the same time, because this is reality in New Zealand".
"This terrible act of cruelty and senseless death of a young kitten is what we have to fight against! We can't even begin understand why this little soul was stoned to death," the post said.
"What the hell is happening with our youth if they can do this and laugh, and in front of at least one young child."
Paw Justice asked people to "please stand with us, identify them".
"Let them know it's not right, and in fact it's so wrong it's beyond sick. Let them know it won't be tolerated."
Paw Justice co-founder Craig Dunn told RNZ the video was "one of the most disturbing videos I've seen ... it's just horrific".
"It was so casual and blasé - there was no empathy, no consideration, nothing towards it."
Dunn told RNZ the people involved "need to be brought to law and made accountable" for the "sadistic" behaviour.
This morning the organisation posted an update, thanking those who had helped to identify the girl in the video but asking people not to name and shame anybody.
"Please do not post addresses or make calls for vigilante justice. We cannot allow this for many reasons, including legal and privacy reasons," the post read.
"We understand the anger, trust us, we do. But we have a job to do, and we have to do it properly."
Anyone with information on the incident was asked to email craig@pawjustice.co.nz with the word "kitten" in the subject line.
People could also text "reward" to 4413 to help offer a reward for information leading to those responsible being caught.
Police say they are aware of the video and are making inquiries.