The paper is concerned with the properties, in particular the finiteness and solvability properties, of the stable group \(\xi(X)\) of homotopy classes of self homotopy equivalences of a connected, finite CW complex \(X\) (i.e. \(\xi(X)\) is isomorphic to the group of homotopy classes of self equivalences of the \(r\)-th reduced suspension of \(X\) for \(r\) sufficiently large). By considering the ring homomorphism \(H\colon [X,X] \to \End H_*(X)\) (where \(H\) denotes the reduced integral homology functor) and using a result of \textit{E. H. Spanier} [Sympos. internac. Topología algebráica, 259--272 (1958; Zbl 0092.15703)] that the additive homomorphism \(H\) is an isomorphism modulo torsion, it is shown that for a connected, finite CW complex \(X\) firstly \(H\colon \xi(X)\to \Aut H_*(X)\) has a finite nilpotent kernel and secondly if \(\beta_i(X)>1\) for some \(i\) then \(\xi(X)\) contains a nonabelian free group. Accordingly if \(\xi(X)\) is infinite it is nonsolvable. In the case where \(\xi(X)\) is finite (i.e. when \(\beta_i(X)\le1\) for all \(i)\) conditions are determined sufficient to ensure the solvability of \(\xi(X)\). Then simple examples are constructed to demonstrate that these conditions cannot be deleted. Note: \textit{D. W. Kahn} has independently undertaken a study of the group \(\xi(X)\) and has constructed an explicit countable family of finitely-generated groups amongst which \(\xi(X)\) must lie [Topology 11, 133--140 (1971; Zbl 0214.21904)].