Let X be a connected CW-complex. The group G_#(X) of all based homotopy classes of self-homotopy-equivalences of (X, *) inducing the identity automorphisms of all homotopy groups is studied, and we obtain the following exact sequence concerning the twisted principal fibration p : <P^^￣>_0 → B with fibre K(G, n)
H^n(B ; G) → G_#(<P^^￣>_0) → G_#(B),
where H^n(B ; G) is the cohomology with the local coefficient induced by qθ: B→ L
= Lφ(G, n+1) → K = K(π_1 (B), 1).