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](https://stacks.math.columbia.edu/tag/09XP)).


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?



Does a morphism of etale sheaves restricting to a closed subscheme $Z$ induce a morphism of their subsheaves of sections supported on $Z$?
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 $i^\*\mathcal{H}\_Z(\mathcal{G})=0$.


