# Does a morphism of etale sheaves restricting to a closed subscheme $Z$ induce a morphism of their subsheaves of sections supported on $Z$?

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?

If I understand the first question correctly, the answer is no. Assume $$Z\neq\emptyset$$ and $$X\smallsetminus Z$$ is dense in $$X$$. Let $$A$$ be any nonzero abelian group and take $$\mathcal{G}=\underline{A}_X$$ (the constant sheaf), and $$\mathcal{F}=i_*(\underline{A}_Z)$$.
We have $$\mathcal{H}_Z(\mathcal{F})=\mathcal{F}$$ and $$\mathcal{H}_Z(\mathcal{G})=0$$. The natural map $$\mathcal{G}\to\mathcal{F}$$ induces an isomorphism $$\psi:i^*\mathcal{G}\to i^*\mathcal{F}$$ (both are canonically isomorphic to $$\underline{A}_Z$$).
Now consider $$\phi:=\psi^{-1}:i^*\mathcal{F}\to i^*\mathcal{G}$$. This is an isomorphism which does not map the subsheaf $$i^*\mathcal{H}_Z(\mathcal{F})=i^*\mathcal{F}$$ into $$i^*\mathcal{H}_Z(\mathcal{G})=0$$.