Macbook Pro Crashing and Startup Failure Firstly, some background information for the problem:

My girlfriend's Macbook Pro had an issue that was annoying, but not fatal. Roughly once a day (but sometimes not for a few days, and sometimes twice or more in one day) the laptop would suddenly freeze and turn itself off. It would sometimes immediately restart itself, and sometimes remain dead until the power was turned back on. Either way, if it was turned back on very soon after the crash, it would more often than not die half way through start-up. If we waited ten or twenty minutes to turn it back on it was usually fine. This never resulted in any lost data and there were no other signs of a problem during normal functioning, so it was never more than a mild inconvenience, until last night that is.

Yesterday, we were trying to fix this or at least understand the problem. We ran an Apple hardware check by holding D during startup, which found nothing. To help eliminate software issues, we updated it to the latest version of OS X. The update went well, and for half a day it was running Yosemite with no noticeable problems. Then, in the evening, the usual crash happened again. However, now the computer does not turn back on. Upon booting, the startup sound plays, and then the Apple logo appears along with a progress bar - much like the progress bar that appeared during the Yosemite installation - which reaches around half completion before going to a blank white screen and turning off after a few minutes. If the laptop is plugged into mains power, it keeps restarting itself over and over again and failing each time. If it is running on battery power, it just dies once and waits to be turned back on.

Being unable to fully start up the computer any more, we don't have access to all the information that might come in handy. I'll list some details that might have some chance of helpful, but I might not remember everything all at once so please do not hesitate to ask questions or for clarification on anything unclear.

-It's a Late 2011, 15-inch Macbook Pro (4GB of RAM, probably the 2.2GHz processor)

-The frequent crashes often happened while playing video files, and (we think) it only happened when using Airplay to mirror the screen through Apple TV. It has happened using two different video playback programs.

-As mentioned in the text, Apple Hardware Check found nothing before. Now, we cannot seem to access Apple Hardware Check - only the OSX-update-like progress bar of doom. We also can't seem to access safe mode with it.

-We tried resetting PRAM, after it wouldn't start up any more. It didn't change anything.

-The last, fatal crash, after which it will not start up any more, happened during Airplay as usual, but also while updating the software on an iPhone through iTunes. The iPhone also died, but was fine after being plugged into another computer with iTunes and following through with the update there.

It was obviously somewhat stupid to not get this problem fixed while it was only an inconvenience, but it was not reproducible at that stage, so our hopes of having someone fix it were low. Now we're trying to get as much information as possible about what could be causing this and how we might be able to go about fixing it. Any help and/or information towards fixing this issue, or advising me on how to go about getting it fixed, is thus appreciated.
Apple has an extended guarantee program for that:
https://www.apple.com/support/macbookpro-videoissues/