With backing from Foundation X, they are the ones responsible for the creation the Gaia Memories, which they give to criminals and other interested parties to use in order to study them.
Kamen Rider W
Through their Gaia Memory Distributors, Museum handed out Gaia Memories to various criminals in order to study their use, seeking to use the data gathered to trigger the Gaia Impact and bind humanity to the Earth.
They were opposed by Kamen Rider Double, who sought to stop them from distributing the dangerous devices that could cause their users to go insane. Their plans would ultimately never come to fruition after Ryubee Sonozaki was defeated and Foundation X cut off their research into Gaia Memories.
Let's Go Kamen Riders
In the altered timeline that emerged from Shocker achieving their goal of global hegemony through Shocker Greeed, Museum was one of the organizations that became a part of Shocker in the new reality, being represented on Shocker's leadership council by the Terror Dopant.