The FAA doesn't have an "avionics" certification. The FCC issues a GROL (General Radio Operators License, big brother of the Restricted Radiotelephone permit that actually requires a test to obtain) which basically just allows you to operate radar systems on the ground, and remove/replace avionics boxes, and many A+Ps have it as many shops still require it. There is an AET certification available from the independent NCATT that is becoming more recognized as the "standard" for avionics technicians.
For the most part, avionics are still handled under rapair stations, just like a prop, the A+P is basically limited to removing it, sending it out, and replacing it when it returns.
Unfortunately, the A+P certificates are still very generalized, with no distinction between working on a Piper Cup and a 777 as far as training and testing requirements. Every A+P must still be able to demonstrate working with fabric wing coverings, wooden airframes, etc.. and not enough emphasis is put on things like NDT, hydraulics and avionics as should be. The system needs an overhaul.
As far as outsourcing, it is a reality that mx is being outsourced in large fashion, and not just to foreign providers. There is a major domestic mx contractor in the US that the Air Force, Southwest, and United have all been sending more and more work to. This particular contractor pays just above minimum wage, has been repeatedly cited for I-9 employment violations, OSHA violations, and various other labor violations. Unfortunately, the rules of cabotage do not apply to mx, and it is also being sent overseas. At the part 147 school I attend, it is a little more than disturbing to see how many foreign nationals are attending school here to return to their home country and take the business with them for mere peanuts.
but ah, that's the new American model: cheaper, faster, on-time and under budget, forget consequences!