Fibered groups with non-trivial centers

Let (G, F) be a fibered group and let Γ be a semigroup of endomorphisms of G such that {id.}? Γ and σ(A) ? A for σ ∈Γ, A ∈F. In this paper we investigate the structure of fibered groups (G, F, Γ) which have a non-trivial center. We start with general considerations and conclude with the construction of a large class of examples. In the situation in which Γ ≠{id.} much structure can be obtained. In particular, when G is a finite group G is a p-group and in many cases also of exponent p.