Retraction $S^2 \vee S^4 \rightarrow S^2$ collapsing $S^4$ to the wedge point induces homomorphism on second cohomology.

Question

Let $r: S^2 \vee S^4 \rightarrow S^2$ be the retraction collapsing $S^4$ to the wedge point. I am trying to show that the induced map $r^{*}: H^{2}(S^2) \rightarrow H^{2}(S^2 \vee S^4)$ on the second cohomology groups is an isomorphism.

We get that it is an injective homomorphism for free because the inclusion map induces a left inverse for $r^{*}$. I need a hint on how to show it is surjective.