Proof:

Let $r: X \longrightarrow A$ be a retraction such that $j\circ r \sim$   id$_{X}$. By (the proof of) theorem 11.6, $r_*$ is injective. But the composition $r \circ j =$   id$_{A}$ shows that $r_*$ is surjective. Hence $r_*$ establishes an isomorphism between $\pi_1(X, x_0)$ and $\pi_1(A, x_0)$.