As pointed out by Mikko Korhonen in this answer, Özdem Çelik proved (in 1976 here) that a finite group whose Sylow subgroups are cyclic (called a Z-group) is determined by its character table.

Now there are many results and conjectures relating character tables and Sylow subgroups (see this paper of Gabriel Navarro), the most famous being perhaps the McKay conjecture.

This leads to wonder whether Çelik's theorem can be extended*.

Question 1: Is a finite group determined by its character table only if its Sylow subgroups are so?
Answer (Alex B.): No.

Question 2: Is a finite group not in a Brauer pair only if its Sylow subgroups are so?
(negative answer suspected by Alex B.)

*Question 3: Is a finite group determined by its character table if its Sylow subgroups are so?
(it is this question which wonders whether Çelik's theorem can be extended)

Question 4: Is a finite group not in a Brauer pair if its Sylow subgroups are so?