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Constable Chris Wakelin, of Ohai, said those responsible were aged between 11 and 16 years old.
"The offenders will all be interviewed and referred to the Invercargill Youth Services team for appropriate action," he said.
Three more kittens were removed from the home of one of the offenders after police worked in conjunction with the SPCA, Const Wakelin said.
"We recognise that the content of the video has been extremely upsetting for people across New Zealand.
"Our staff have also been sickened by the video."
Meanwhile, Netsafe is urging people to refrain from sharing other people's personal content online, after the video of the kitten being harmed spurred death threats to those inviolved.
Animal rights group Paw Justice posted it to Facebook last night.
While Netsafe agreed the actions captured in the footage was horrific, chief executive Martin Cocker believed sharing the video had put a person's life in danger.
"People shouldn't be exposed to that kind of content. The actions shown in the video should never have been undertaken either but the video shouldn't have been shared," he said.
While Mr Cocker said it was understandable that Paw Justice was outraged at what the video showed, he suggested the best response was to report it.
"If you come across that kind of content you can report it to police or to Netsafe. You cross a line when you identify anyone involved in something like that, or make threats towards them."
Paw Justice had initially asked people to "please stand with us, identify them", however a second post this morning asked 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."
- ODT online and NZ Herald