Let $X$ be a locally Noetherian scheme and $i:Z\to X$ be an immersion of closed subschemes. Let $\mathcal{F},\mathcal{G}$ be two etale abelian sheaves over $X_{et}$.

We can define the subsheaf $\mathcal{H}_Z(\mathcal{F})\subset \mathcal{F}$ of sections supported over $Z$, i.e. for any etale morphism $h:V\to X$ $$\mathcal{H}_Z(\mathcal{F})(V):=\{s\in\mathcal{F}(V)|\text{ Supp}(s)\subset h^{-1}(Z)\}$$ (check out this tag on stacks project).

Assume that we have a morphism of etale abelian sheaves over $Z_{et}$ $$\phi:i^* \mathcal{F}\to i^* \mathcal{G}$$ Can we induces a map $i^*\mathcal{H}_Z(\mathcal{F})\to i^*\mathcal{H}_Z(\mathcal{G})$ over $Z_{et}$?

If so, does it naturally follow that a map $i^*\mathcal{F}\to i^*\mathcal{G}$ can induce a map $\text{H}^n_Z(X_{et},\mathcal{F})\to \text{H}^n_Z(X_{et},\mathcal{G})$ which is my ultimate goal, and can the argument safely transfer to fppf sheaves?