In this paper we study the relationship between the extended symmetries of exact Courant algebroids over a manifold M defined in [1] and the Poisson algebras of admissible functions associated to twisted Dirac structures when acted by Lie groups.We show that the usual homomorphisms of Lie algebras between the algebras of infinitesimal symmetries of the action, vector fields on the manifold and the Poisson algebra of observables, appearing in symplectic geometry, generalize to natural maps of Leibniz algebras induced both by the extended action and compatible moment maps associated to it in the context of twisted Dirac structures.