price of admission vs fee

price of admission and fee both are nouns.

Word	Noun	Adjective	Verb	Adverb
price of admission	Yes	No	No	No
fee	Yes	No	Yes	No

As nouns, fee is a hypernym of price of admission; that is, fee is a word with a broader meaning than price of admission:

price of admission (noun)	fee (noun)
the fee charged for admission	a fixed charge for a privilege or for professional services
	an interest in land capable of being inherited

price of admission (verb)	fee (verb)
	give a tip or gratuity to in return for a service, beyond the compensation agreed on