An additive group semiautomaton or, in brief, GS-automaton is a generalization of the well known linear state machines. The purpose of this paper is to study the near-ring of endo-transition preserving functions of additive GS-automata. This class of near-rings is a subclass of the celebrated centralizer near-rings, and includes near-rings of infra-endomorphisms. Complete characterizations using both algebraic and graphical properties of the additive GS-automaton such that the near-ring is $0$-symmetric or constant are given. Conditions such that this near-ring being simple or being a ring are also provided.