  We consider Group Control by Adding Individuals (GCAI) in the setting of
group identification for two procedural rules -- the consensus-start-respecting
rule and the liberal-start-respecting rule. It is known that GCAI for both
rules are NP-hard, but whether they are fixed-parameter tractable with respect
to the number of distinguished individuals remained open. We resolve both open
problems in the affirmative. In addition, we strengthen the NP-hardness of GCAI
by showing that, with respect to the natural parameter the number of added
individuals, GCAI for both rules are W[2]-hard. Notably, the W[2]-hardness for
the liberal-start-respecting rule holds even when restricted to a very special
case where the qualifications of individuals satisfy the so-called consecutive
ones property. However, for the consensus-start-respecting rule, the problem
becomes polynomial-time solvable in this special case. We also study a dual
restriction where the disqualifications of individuals fulfill the consecutive
ones property, and show that under this restriction GCAI for both rules turn
out to be polynomial-time solvable. Our reductions for showing W[2]-hardness
also imply several lower bounds concerning kernelization and exact algorithms.
