  We solve the seventh problem of Oliver's list [M.\ Aschbacher, R.\ Kessar,
B.\ Oliver, \textit{Fusion systems in algebra and 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.
