When you get to the third island, Tan Line Island, take the top route past the scorpion girl to reach the temple. Talk to Squid Baron to obtain the X-Ray Specs. Now you need to find three Mummy Statues to obtain their designs. They are located:
1) Tan Line Island. Go back to the scorpion girl enemy, then take the lower route. The first mummy is directly ahead of you.
2) Scuttle Island. Go to Sky's house, the one with the bird eggs who loaned you the library card earlier. After the cutscene, the second mummy is here.
3) Saliva Island. Backtrack to the platform statues and hit them in the same order to ascend, 4312. Once you're to the chains, climb to the head of the statues, then take a running leap and use the hat to reach the exit on the far left. Follow this path up and around for the final mummy.
Once you've obtained all three, you'll gain the password for the temple door and the achievement.