On the finiteness of the subgroup of homotopy self-equivalences inducing the identity on homology

For a rationally elliptic space Formula: see text, we denote its group of self-homotopy equivalences as Formula: see text, and its subgroup consisting of elements inducing the identity on the homology groups as Formula: see text. This paper aims to investigate how the homotopy groups of Formula: see text influence the finiteness of Formula: see text.