Up until 1888 Rarotonga was an independent Kingdom. It became a British Protectorate in 1888, along with the other islands in the northern and southern groups of the Cook Islands, due largely to fears that France might occupy the territory as it had Tahiti. Royalty continued to rule at a local level.
Stamps inscribed "Cook Islands Federation" were issued then, despite the local government being anything but a unified federation.
In 1901, Britain and New Zealand conspired to let New Zealand annex the group despite opposition from the country's traditional chiefs. As many of the islands were independent and locally ruled by kings and chiefs, they had little unity to control the events.
To make a long story short, the chief of Tongareva (Penrhyn) in the northern group, and the king of Aitutaki and environs objected the use of the issued Cook Islands stamps bearing the Queen of Rarotonga, as she had no rule over those islands.
As a result, in 1919, Cook Island stamps were replaced by New Zealand stamps overprinted for use in Rarotonga, Aitutaki and Penrhyn, followed by regular stamps inscribed with the same names.
By 1932 stamps inscribed Cook islands were once again used throughout the territory and continue as today.
Queen Makea of Rarotonga (only), NOT queen of all the Cook Islands