From catalogs Robert has posted, Gamewell seemingly was part of the Cerberus group starting around 1984 edit:1982 to, and according to a now defunct history page, was purchased by Honeywell in 2003. I could not find any information on when they left the group some time before or after Cerberus Pyrotronics was bought by Siemens, or if Honeywell purchased Gamewell from Siemens.
We also know that Cerberus Pyrotronics has rebranded the Century series of pulls, which most likely happened during that time.
Given that a lot of Gamewell’s panels during that time were rebranded Cerberus Pyrotronics panels [1] [2], the latter rebranding the former’s pull stations, and that Siemens kept around the Faraday name for a while after purchasing it makes it hard to tell whether the Gamewell name would be kept around because it’s recognizable, or whether it would be slowly folded into the Siemens name like Cerberus Pyrotronics and Faraday.