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Key and MediaWorks were heavily criticised for the stunt at the time, with calls made for Key to be removed as an ambassador for White Ribbon, a charity against violence.
MediaWorks said at the time comic material will often refer to challenging concepts, but considered that "drawing humour from unsavoury ideas is not the same as trivialising or condoning them".
It said that there were no direct or explicit references to sexual violence but rather "euphemisms reliant on well-worn cliches that required a certain level of maturity among listeners to be understood".
The independent broadcaster said the material in question did not fall outside what regular listeners of both the station and the programme would expect, and that sufficient care was taken to ensure the content was acceptable in context.
Meanwhile, Key's office said: "The Prime Minister does these interviews in the spirit of Christmas and the content is decided by hosts.
"The interviews are meant to be light hearted, and the Prime Minister hopes the media and the public take them that way."
However, BSA deemed that the stunt was a deliberate reference to prison rape and trivialised sexual violence.
In its decision, the authority said: "Sexual violence is a serious issue which affects some of the most vulnerable people in society, including those who are incarcerated."
It found that the content went "beyond currently accepted norms of good taste and decency into something that was inappropriate and in poor taste, and would have offended many people".
"This was not an off-the-cuff or fleeting joke made on the spur of the moment. Rather, the gathering and use of props such as soap, as well as quoting from the film Deliverance, in our view clearly required some forethought."
The authority noted that while The Rock and The Rock Morning Rumble were known for an irreverent and controversial brand of humour, this broadcast went too far and showed poor judgment.
The authority ordered MediaWorks to broadcast a statement summarising the upheld aspects of its decision on top of costs to the Crown.
A MediaWorks spokesman said it accepted the authority's decision. He wouldn't divulge when the hosts would apologise on air and refused to say if they would face any disciplinary action.
A spokeswoman for the Prime Minister said: "As the Prime Minister has said, he didn't know what was going to happen and didn't pick up on the connotation at the time.
"We have nothing further to add."