I doubt she had any idea that the third group even existed. Not sure what to think, except her response must not have been to their liking.
Sucks to be her.
At the time she filled in the form, she was obviously aware the third group existed as she had written to and visited one of its members in prison.
It is fairly obvious that her relationship with the "terrorist" organisation was very tenuous, but one point of a background check is a test of your willingness to be full and open about your past. In fact if she had given a full open answer, I suspect there would not have been a problem.
Despite conspiracy theories to the contrary, government agencies do not know everything about you. A background check will not necessarily find out everything about your past, but if it detects evasive answers then it is grounds for not employing someone in case there is more the potential employee is not telling or deliberately hiding.