Is it typically the role of an oral surgeon to attach the abutment?

I'm exploring the option of dental implants to replace a couple of irreparably damaged teeth. I've learned that the dental implant process involves three components: the implant itself, the abutment, and the crown. It's clear that if I choose an oral surgeon for the procedure, they will handle the implant surgery, and my dentist will take care of the crown. However, I'm unsure about who typically manages the abutment: the surgeon or the dentist?