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

Question

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?

Answer 1

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 $i^*\mathcal{H}_Z(\mathcal{G})=0$.