I can't make heads or tails of whether Idea was the lab that ran that project or if they're still around. They're involved in Indian Poker?
Ideal was the lab from the flashbacks. It ran the experiment with Ghost-chan and was associated with the lab the developed Misaki's powers. Their goal seemed to be to use Ghost-chan's powers to create a brainwashing card that could also be used to develop powers with AIM mumbo jumbo. Basically it seems similar to the Level Upper from Railgun's first arc.
They're no longer around, but the survivors of the experiment, Stitches and Ayu, are carrying on the experiment and, in Ayu's case, perhaps trying to get revenge on someone for the experiment. I recall that Ayu had beef with Misaki in the novels so she may blame her for the experiment going wrong. Something like "this experiment never would have even happened if Misaki wasn't there to influence the lab to use brainwashing."