Francesco,
By "group" do you mean what Moodle calls a cohort?
If so, there is no way to do that directly. Moodle does not currently have the ability to assign users in cohorts or groups like that to roles. A role can only be assigned to an individual. So, you must assign those users the new role individually in the Assign system roles tool.
That will work if the list is small.
If you have a lot of users, hundreds or thousands to add, you could also do this using the User
upload tool with a csv file to assign a system role in bulk. How to do that is explained here:
https://docs.moodle.org/310/en/Upload_users#Set_system_roles. If you already have a list of the users in that group in a spreadsheet file, then you could do it that way.