As pointed out by Mikko Korhonen in [this answer][1], Özdem Çelik proved (in 1976 [here][2]) that a finite group whose Sylow subgroups are cyclic (called a [Z-group][3]) is determined by its character table.    

Now there are many results and conjectures relating character tables and Sylow subgroups (see [this paper][4] of Gabriel Navarro), the most famous being perhaps the [McKay conjecture][5].  

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?  



  [1]: https://mathoverflow.net/a/326735/34538
  [2]: https://mathscinet.ams.org/mathscinet-getitem?mr=0470050
  [3]: https://en.wikipedia.org/wiki/Z-group
  [4]: https://doi.org/10.1007/978-981-13-2047-7_10
  [5]: https://en.wikipedia.org/wiki/McKay_conjecture