MBP 15" Late 2011 rebooting

Try running the Apple Diagnostics to see if any hardware issues are detected.

The three most likely causes in no particular order (there may be other causes too):

• Third party software issue -- Safe Mode as already suggested should tell us this
• Failing hard drive
• GPU failure -- Safe Mode may work depending on the severity of the GPU failure (GPU failure very common on the 15" model).

Try to boot the Mac up in the Safe mode to see if the problem persists

Start up your Mac in safe mode – Apple Support (UK)

You can also try and press CMD+R on startup to see if you can boot in the macOS recovery menu to install macOS from there

If you have a Time Machine backup you can put that one back in the install procedure

Download the Knoppix DVD .iso file with "EN" in the name for ENglish. Either the v8.6.1 or v9.x should be fine. Use the downloaded Knoppix .iso file as a source for Etcher (Mac/Windows/Linux). Option Boot the Knoppix USB stick and select the orange icon labeled "EFI". While Knoppix is booting the Mac may appear to be frozen on the Apple boot picker menu so make sure to give Knoppix lots of time to finish booting.

If Knoppix boots to the desktop, then click the "Start" menu on the lower left corner of the Taskbar and navigate the menus to "System Tools --> GSmartControl". Within the GSmartControl app double-click on the drive icon for the internal drive to access the drive's health report. Post the complete health report here. Also check the "Error" tab within GSmartControl to see if there are any errors listed and post them here as well.

If Knoppix just boots into a command prompt instead, then let me know as I can provide instructions for checking the drive's health using the command line as well. I find that Knoppix tends to boot into the command line instead of the GUI when there is an issue with the discrete GPU.

Since you managed to boot to the Knoppix desktop, just follow my first set of instructions to use GSmartControl in this post:

MBP 15" Late 2011 rebooting - Apple Community

It is not worthwhile to replace the GPU as it is an expensive repair either requiring replacement of the whole Logic Board (potentially the replacement will have the same issue), or finding a repair shop with the expensive specialized equipment to replace the GPU. FYI, Here is what is involved:

https://www.youtube.com/watch?v=U9-9EpilsBw

On a Mac, booting into Safe Mode should utilize a more basic video driver which may allow the laptop to boot even with a GPU issue, however, some of the GPU issues associated with these Apple laptops will affect the laptop in other ways beyond just video...I have seen it prevent the laptop from even powering on (at the time Apple replaced the Logic Board for free under the free GPU repair program that was active at the time..now expired). Unfortunately a 2011 laptop is considered "Obsolete" by Apple, so Apple will no longer service this laptop even for a paid repair. Unless the problem is due to the internal drive or memory, it is time to retire this laptop.

A third party battery may fail the diagnostics even if the third party battery is healthy. So the diagnostics haven't revealed anything useful. Unfortunately the diagnostics don't detect many drive failures so a failing hard drive is still a real possibility. If you want to check the health of the hard drive, then I can provide instructions for creating and using a bootable Linux USB stick, just let me know if you are interested in this option.

Are you saying you don't see the orange icon labeled "EFI" when hold the Option key immediately after the startup chime while the Knoppix USB stick is connected?

Or were you able to select the orange icon labeled "EFI" to begin booting Knoppix and it ends up with a blue screen? If you are able to do this, then lets try to force Knoppix to boot to the command line instead of the graphical desktop environment. I have seen Knoppix fail to boot to the desktop on systems with GPU issues.

Trying to force Knoppix to boot to the command line is a little tricky since you need to type something on the screen without being able to see what you are typing. Once you Option Boot the Knoppix USB stick, wait about 10 or 15 seconds, then press the F2 key. Then type the following all in lower case (make sure there is at least one space before the "2"...I put several just to accentuate the spacing so it is seen easily):

knoppix64   2

Press the "Return" key at the end of the line to execute the command. You may not see anything happen for a little bit, but hopefully you will at some point see a boot log or the command prompt.

If Knoppix boots to the command line, then you can issue the following command to list the connected drives (make sure there are no other external devices connected except for the Knoppix USB stick when booting Knoppix). We need to verify the drive identifier for the internal Mac drive.

sudo  lsblk  -f  |  grep  -i  sd

This should just display the internal and external drives. If there is only one internal drive and the Knoppix USB stick, then you will (in theory) just see two different drives listed identified by "sda" and "sdb" (there will be items linked from them for each partition on the respective drive designated by a trailing number -- the numbers are irrelevant for our purposes). One of these should be clearly listed as "Knoppix" while the other one may or may not show as Mac related. We need the drive identifier for the Mac drive which I believe will be designated as "sda", but it has been a while since I've done this myself.

Use the drive identifier for the Mac drive (or the drive identifier which is not listed as Knoppix) in the following command. If the Mac drive is instead associated with the "sdb" entries, then replace "sda" in the following command with "sdb" or whatever drive identifier is associated with the Mac drive.

sudo  smartctl  -a  -s  on  /dev/sda  |  less

When you run this command, hopefully it will show the health report for the internal Mac drive. The " | less" part of the command actually will stop the output from scrolling off the screen. Just use the up & down arrow keys to scroll up & down the report so you can take a pictures of it to post here. Once you scroll to the end of the report and have taken the pictures and are done, you can just press the "q" key (lower case "Q"). You will be returned to the command prompt. If there is an error message or nothing is shown, then you may have used the wrong drive identifier in the command (you may still need to press "q" to get back to the command line).

Then just type the following to power off the Mac:

sudo  shutdown  -h  now