The authors study well-posed exactly (approximately) controllable and exactly (approximately) observable continuous-time linear system realizations of scalar-valued inner functions on the open right half-plane. They give necessary and sufficient conditions for the inner function \(G\) to have such a realization with exponentially stable associated \(C_0\)-semigroup. The question of when \(G\) has such a realization with its associated \(C_0\)-semigroup being a group is answered in terms of the Carleson measure constructed by zeros of \(G\). The authors' results on exponential stability turn out to contradict the results of \textit{R. Ober} and \textit{Y. Wu} [SIAM J. Control Optimization 34, 757-812 (1996; Zbl 0856.93051)], and they show where Ober and Wu made a mistake and how this may be repaired. Finally, a necessary condition is derived for \(G\) to have a realization whose associated \(C_0\)-semigroup is a \(C_0\)-group and which is exactly controllable in finite time.