Does a deformation retraction of $X$ onto a subspace $A\subset X$ induce an isomorphism $\pi_n(X) \to \pi_n(A)$?

Question

Let's say we have a topological space $X$ and a subspace $A\subset X$. Assume $A$ is a deformation retraction of $X$. Does that imply that the induced homomorphism of the deformation retraction is an isomorphism on all $\pi_n$?

All i could find was the usual proof regarding the induced homomorphism of the inklusion $i$ on the fundamentalgroups of $X$ and $A$ respectively.