What You Will Need
In order to complete this Easter Egg, you need two previous achievements from Call of the Dead and Shangri-La, which are "Ensemble Cast" and "Time Will Tell". When those achievements are earned, the player who incarnates Richtofen in Moon will see a tiny Vrill Generator icon on the bottom of his screen.
For this Easter Egg you'll be requiring three weapons from the Mystery Box: the Wave Gun, Quantum Entanglement Devices(QED), and Gersch Devices. You'll also need the Excavator "Pi" to breach into Tunnel 6. It does not matter when it breaches, it only needs to breach once.

Part 1

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.

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.

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.

Step 4
When tunnel 6 has been breached, have someone hack the proper terminal from the Receiving Area so that the Digger leaves.

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.

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.

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.

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.

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!

Part 2

Step 9
Head back to the "No Man's Land" (Area 51 where you began at). To your right stands a beam structure with some plates stores on it. Throw a grenade & the plates will fall off.

Step 10
Have someone throw a Gersch Device behind the crates, right where the plates have fallen. If done properly, the plates should now be on the Teleporter Pad.

Step 11
Go back onto the Moon. The plates should now be right in front of the Quick Revive. Have someone use a QED on them, after doing this they should now be right next to a computer in a corner of the same room.

Step 12
Look in the labs for a lead pipe. It is often hidden in one of the counters where the hacker spawns. Grab it and plug it into the machine back in the Receiving Area. Have Richtofen put the Vrill Generator in between.

Step 13
Have Richtofen repeatedly mash the X button on the computer keyboard for a few minutes. Wait until the voice over ends, the computer screen will STAY green.

Step 14
The Vrill Generator will now be powered up and shiny. Grab it and head back to the pyramid. Get near the engraving and put the Vrill Generator in it. There will be four tubular containers rising up from the corners of the pyramid base.

Step 15
Fill up the four tanks by killing around 25 zombies near each container. The same sound emits when a container is full.

Step 16
Activate the switch next to the pyramid again and the containers will be drained of their content. Have Richtofen grab the Vrill from the hole in the engraving upfront. Player Richtofen now earned permanent perks!

Step 17
Head back to the four "Samantha Says" consoles where you need to play three increasingly more difficult rounds.

Step 18
Time to head back to the power room. Have someone use a QED on the engraving to move the black egg again. (Tip: Go prone and get as close as possible to the engraving, then aim low at the golden ring then throw the QED).

Step 19
The Egg will fly back to the outside area, right next to the "Samantha Says" consoles. Use a Gersch Device on the egg.

Step 20
You'll see the Egg teleport onto three rockets outside of the playable area where they'll launch and head towards the Earth. The final achievement "Big Bang Theory" will unlock and all players will earn permanent perks. The Moon is now yours!