General managers can take different forms around the league, but by and large, they tend to be the single person most responsible for putting together both the 90-man roster in the offseason and turning that into a 53-man roster in the regular season—as well as constantly maintaining it in between, and at every other point.
Even teams with head coaches who are prominent decision-makers such as the Pittsburgh Steelers and Mike Tomlin recognize that at some point, the keys turn over to the general manager. While the team is in-season, the head coach is…head coaching.