The People Picker, as triggered from the Group Member management page (/registry-pe/group-members?group_id=x), appears to generate problematic URLs and/or queries. Specifically, a query looks something like

/registry-pe/api/ajax/v2/people?co_id=1&picker=on&for=GroupMembers&extended=PrimaryName%2CIdentifiers%2CEmailAddresses&identifier=hunt&mail=hunt&given=hunt&family=hunt&groupid=1&action=index&limit=7&page=1&direction=desc

In summary (more details in the comments) and in decreasing order of importance:

1. Specifying multiple fields (identifier, mail, given, family) results in an AND query that does not correctly search the database.
2. co_id and groupid are conflicting.
3. The cake action (index) and the flag action=index are duplicative, and the use of "picker=on" is semantically conflicting.
4. groupid does not follow the pattern of naming parameters according to the actual database field (ie: group_id).