This quest is started by looting the dismembered body of Private Hart in the western yard of the Museum of Witchcraft
. Listen to her holotape then enter the museum through the basement door nearby.
There will be constant shaking and you'll immediately feel you've stepped in something out of a horror movie. Continue moving along the basement and loot any containers and bodies you'll find. Continue upstairs and carefully walk around. You'll have to fight a very high-leveled Deathclaw so it will be a good idea to lay down traps in advance. Or, if you're really confident, just launch two mini-nukes at it (from a safe distance of course)
There's a copy of Grognak the Barbarian on the table in the corner of the upper floor and a green steamer trunk nearby. Find the mutilated body of Sgt. Lee and play his holotape. Also, collect the cracked deathclaw eggs and the single pristine egg near the body.
You'll now have two options: return the egg to its nest east of Recon Bunker Theta or complete the delivery in Diamond City. If you deliver the egg, the non-hostile mama deathclaw will appear and the mission will be complete.