Step 1
When the power has been activated, you'll need to play a round of "Samantha Says" on the four computers in the outside area near the beginning.
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Step 2
Next, work your way to the labs. Here search for the "Hacker" nearby on one of the counters. Using the hacker, hack either one of the four red buttons along the wall. This will activate four switches nearby as well. You can locate them by searching for their green light that should be on by now.
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Step 3
Now you need to hack all four of these switches. Once this has been accomplished, the four buttons you found on the wall earlier must be pressed in less than 2-3 seconds. When done properly, they will stay lit. This will open up the round hole in the pyramid front graving. Of course you'll be needing this for another step.
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Step 4
When tunnel 6 has been breached, have someone hack the proper terminal from the Receiving Area so that the Digger leaves.
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Step 5
Go back safely to tunnel 6 and see that a Black Egg has appeared near the M16. Knife it and you'll see it moving around the room. Follow it since it will get stuck in various places. Knife it again and open doors for it until it flies up onto a dish on top of the spatial base.
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Step 6
Have someone use the Wave Gun on the tip of the Dish. This will cause the Egg to move again. Follow it through tunnel 11. It will go hide into some metal railing at some point. You can either shoot it from the wall next to Stamin-Up of throw a grenade at it. Keep moving it until it reaches the pyramid in the power room. It will fit in the hole you just opened.
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Step 7
A tubular container will rise up from the right side of the pyramid. Next step consists in filling up this container with Zombie Souls. Kill around 25 zombies near the container. You'll see the souls flying up to the ceiling structure which will fill up the tank.
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Step 8
Once filled up entirely, you'll hear a sound. Activate the switch right next to the container; this will open the pyramid, revealing Samantha.
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At this point you will be awarded a Death Machine for 90 seconds. As well as the achievement "Cryogenic Slumber Party" unlocks.
This is only the first part of the Easter Egg. There is still another achievement to unlock! Starting from here is the part where you absolutely need Richtofen's Vrill Generator to proceed!
CREDIT TO A_Lychee_Tree & bombsta101 on TTG