Turn on the power / enable Pack-A-Punch
Head to the Observatory Dome. Use the device called the Harmonic Oculus. You must gather two missing components (found in other areas) and install them. After you start the power sequence, defend the Oculus from zombies while it finishes. Only then will Pack-A-Punch unlock.

Complete the planet-model puzzle
At some point, you’ll obtain a planet model from a trapdoor near the top floor of Machina Astralis. Place it on a rod in that area. Then use the compass directions at the base of the globe light. After that, go back to a brain canister and interact to trigger a lockdown and portal spawn. Survive zombie waves and protect the portal from UFO projectiles.

Go to Mars and continue tasks
Once portalled, you spawn on Mars. Pick up the brain canister that drops, and carry it back to a machine near your Mars spawn. Insert it. This readies the area for boss summoning. 

Fight the final boss: Caltheris
The boss fight has four phases:

• Phase 1: Caltheris appears as a floating skull that fires lasers. Kill zombies to charge generators, then use generators to fire lasers at the boss.

• Phase 2: Boss becomes a rock monster. Shoot its glowing weak spot while dodging its attacks.

• Phase 3: Boss becomes more aggressive and is armored. Break its armor first, then attack weak spots.

• Phase 4: Boss gains new attacks: large rock projectiles and toxic gas clouds. Keep your distance, avoid gas, and focus on vulnerable spots after breaking armor.

After you deal enough damage through all phases, Caltheris dies, and you complete the Easter egg.