Aegis = Indian company providing support services to American OEMs. Therefore, outsourcing is still from America to India and not vice versa. Aegis hiring native Americans for the job is not the same thing as "India outsourcing to America" because native accent is not a requirement/demand of India. It is the demand of OEM consumers of America and thus it implies only that the American jobs have "stayed back" in Americas. There is absolutely no cross-border outsourcing happening at all, whatever be the consideration.
"Native Americans"... you keep using that phrase. I do not think it means what you think it means.
So you are saying cost is the reason to outsource and because of cost that Aegis is hiring US workers to avoid paying for English training, which means Aegis is outsourcing English speakers to the U.S.
The U.S. outsourced call centers and India is outsourcing English speakers. Same thing. Indian companies (Aegis) are hiring Americans = outsourcing = American companies hiring Indians.
I'm not sure why you think it's different.