Brouwer's fixed-point theorem

Question

I'm trying to understand the cohomological proof of this theorem, but I'm stuck at a point, which brings me to ask a more general question: let $i:Z\rightarrow X$ be the inclusion of a closed set into a topological space. If there exists a continuous map $r$ such that $r\circ i=1$, is it true that the induced map on sheaf cohomology groups $$i^*:H^n(X,\mathbb{Z})\longrightarrow H^n(Z,i^* \mathbb{Z})$$is surjective? Why?