Freely homotopic maps from a space to itself

Question

Are two maps from a path connected space to itself inducing the same automorphism on the fundamental group freely homotopic?

Answer 1

Not necessarily. For example, every map $f : S^2 \to S^2$ induces the same map on $\pi_1(S^2) = 0$ but $[S^2, S^2] \cong \mathbb{Z}$.