To update a group name:

groovy:
def someGroup = system.groupManager.get("group name");
someGroup.setName("new name");
system.groupManager.save(someGroup);

You do not need to update authorized configurations when changing group name, as authorizations are associated with group id which will not affected by group renaming.

I have such a system that if I rename group name then I have to also update the configuration path in group authorization

Like if in authorizations the path was
A/B/C/OLD_NAME

then if I rename the group name to NEW_NAME the corresponding authorization path should also be renamed like below
A/B/C/NEW_NAME

Then it can be done via below script:

groovy:
import com.pmease.quickbuild.persistence.*;

SessionManager.openSession();
try {
  def group = system.groupManager.get("some group");
  for (authorization in group.authorizations) {
    authorization.configuration.setName("new name");
    system.configurationManager.save(authorization.configuration);
  }
} finally {
  SessionManager.closeSession();
}

system.configurationManager.save(authorization.configuration);
what is this doing????

Since you also want to change name of related authorized configurations of the group, this statement saves the configuration after changing its name.