The ring of stable homotopy classes of self-maps of A_n^2-polyhedra

We raise the problem of realisability of rings as {X,X} the ring of stable homotopy classes of self-maps of a space X. By focusing on A_n^2-polyhedra, we show that the direct sum of three endomorphism rings of abelian groups, one of which must be free, is realisable as {X,X} modulo the acyclic maps. We also show that 𝔽_p^3 is not realisable in the setting of finite type A_n^2-polyhedra, for p any prime.