Boundary inclusion for a punctured genus-$2$ surface and obstruction to a retraction

Let \(X\) be the closed orientable genus-\(2\) surface with the interior of a small disk removed, and let \(A\) be the boundary circle of that disk.

Calculate the homomorphisms induced by the inclusion \[ A\hookrightarrow X \] on the fundamental group and on homology. Use these maps to prove that \(A\) is not a retract of \(X\).