According to my understanding of US federal OSHA, (CFR1910.178) it is the employer's responsibility to certify the operators of the powered industrial trucks in their facility, and the training and certification is required to be not only "truck specific" but "site specific", thereby documenting and insuring the operator is aware of the hazards of that location's operation.
I like to offer that the word "License" does not apply to powered industrial trucks, the accurate word is "certified".
So then, I guess as to your question of, "Does a 2A Hoisting License cover forklift?" the answer is; No, not really.