Enter, introduce, admit are comparable when they mean to cause or permit to go in or get in.
Enter, in its causative sense, is used chiefly in idiomatic phrases, though occasionally it is employed in the sense to drive or force in.
In idiomatic use it commonly implies writing down (as in a list, a roll, a catalogue, or a record), but in some of these phrases it also connotes the observance of other formalities; thus, to enter a word in a dictionary is to list it in alphabetical order and define its meaning; to enter one’s son at a private school is to send in his name as a candidate for admission; to enter a judgment is to put it upon record in the proper legal form and order.
Introduce is often preferred to enter when it implies insertion. It is the precise word when used of things not native and brought into a country or locality for the first time. Sometimes its use connotes an alien character in that whose entrance is effected.
Admit in this relation usually means let in; it may imply a human agent, but often a means of getting or passing in.