Mr Kennedy has now endorsed Donald Trump forthe presidency, and has been offered a place on the Republican candidate’s transition team if he is elected.
The incident with the whale’s head, which dates to around 1994, was recounted by Kathleen “Kick” Kennedy in a 2012 interview with Town & Country magazine.
Mr Kennedy, who is said to enjoy studying animal skulls and skeletons, reportedly heard that the dead animal had washed up in Hyannis Port, Massachusetts, where the family was staying.
He went to the beach to cut off the head with a chainsaw, and then proceeded to tie it to the roof of the family minivan with bungee cords. The family then drove back to their home in New York, Ms Kennedy said.
“Every time we accelerated on the highway, whale juice would pour into the windows of the car, and it was the rankest thing on the planet,” Ms Kennedy told the magazine.
“We all had plastic bags over our heads with mouth holes cut out, and people on the highway were giving us the finger, but that was just normal day-to-day stuff for us.”
The call for an investigation was made by the Center for Biological Diversity Action Fund, which has publicly endorsed Democratic candidate Kamala Harris for president.
The group wrote a letter to set out their case to marine protection officials at the National Oceanic and Atmospheric Administration (Noaa). The BBC has approached the Noaa for comment.
The letter, quoted by US media, said Mr Kennedy may have broken multiple federal laws by collecting wildlife body parts and transporting them across state lines.
“Vital research opportunities are lost when individuals scavenge a wildlife carcass and interfere with the work of scientists,” the group’s political director Brett Hartl wrote.
“This is particularly true of marine mammals, which are some of the most difficult wildlife species in the world to study,” Mr Hartl added.
Noaa is yet to publicly acknowledge receipt of the letter or issue a comment, and nor has Mr Kennedy. The BBC has approached his campaign for comment.
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