Since its introduction by Symons, the semigroup of maps with restricted range has been studied in the context of transformations on a set, or of linear maps on a vector space. Sets and vector spaces being particular examples of independence algebras, a natural question that arises is whether by taking the semigroup T(mathscr {A}, mathscr {B}) of all endomorphisms of an independence algebra mathscr {A} whose image lie in a subalgebra mathscr {B} , one can obtain corresponding results as in the cases of sets and vector spaces. In this paper, we put under a common framework the research from Sanwong, Sommanee, Sullivan, Mendes-Gonçalves and all their predecessors. We describe Green’s relations as well as the ideals of T(mathscr {A}, mathscr {B}) following their lead. We then take a new direction, completely describing all of the extended Green’s relations on T(mathscr {A}, mathscr {B}) . We make no restriction on the dimension of our algebras as the results in the finite and infinite dimensional cases generally take the same form.