The Group of Homotopy Self-equivalences Inducing the Identity on the Homotopy Groups of an $F_{0}$-Space is Finite

Let $X$ be an $F_{0}$-space and let $\mathcal{E}_{\#}(X) $ the subgroup of homotopy classes of homotopy self-equivalences of $X$ inducing the identity on $\pi_*(X)$. The aim of this paper is to prove that $\mathcal{E}_{\#}(X) $ is finite