I had the same issue on the Galaxy Nexus with Ice Cream Sandwich and have been going nuts trying to figure it out. This info helped except I had to dig to figure out the actual process. Here's how I got it to work.
Yes, you do need to transfer the phone contacts. I went to my contacts list and hit the settings button and selected Import/Export. I then selected Export to storage. Waited a second... then went over to the Groups page.
Again, after hitting the settings button I selected Import/Export - Import from storage.
Now I can type a contact's name into a group in the edit mode and it pops up!
Strange work around, but it worked for me.
Hope it helps.