Signalizer functors, existence, and applications to the fundamental group

We solve the seventh problem of Oliver’s list [M.\ Aschbacher, R.\ Kessar, B.\ Oliver, $\mathit{\text{Fusion}}\mathit{\text{systems}}\mathit{\text{in}}\mathit{\text{algebra}}\mathit{\text{and}}\mathit{\text{topology}}$, LMS Lecture Note Series: 31, Cambridge University Press, 2011] via an explicit signalizer functor construction in the sense of Aschbacher-Chermak for various group models. Moreover we prove the existence of centric linking systems via group models in certain cases which is the first problem and give applications to the fundamental group which is the eighth problem of the list respectively. We illustrate with many examples.