You don't actually "transfer" abilities. The hunter has a chance to learn a pet's ability whenever it uses it, and can then teach that ability to any new pets.
Example: you tame a bear that has Claw 2. After seeing it use Claw 2 enough times, you eventually see how it's done, and can teach it to your cat, and to any other pets you tame in the future (assuming they CAN use Claw, of course).