Do aircraft automatically send a message to the ground if there is a catastrophic failure?
There is nothing formally standardized as far as I know however newer Rolls Royce jet engines (and presumably those from other major manufacturers) do have a continuous inflight data link that reports engine data in real time to their reporting centers. You can read up on it here as well as this discussion with lots more info. 


  Aerospace Operations Centres are able to track the health of thousands
  of engines operating worldwide by using onboard sensors and live
  satellite feeds. Rolls-Royce are world leaders at equipment health
  management (EHM)


The centers are potentially able to detect certain emergencies remotely based on the parameters reported.



Since you note the pilot making contact presumably with the radio: Pilots can report an emergency without the use of a radio by Squawking 7700 on the transponder as well.