I'm trying to show that the sheaf $\mathcal{H}_Z^0(\mathcal{F})$ defined as $\mathcal{H}_Z^0(\mathcal{F}) = (U \mapsto \Gamma_{Z\cap U}(U,\mathcal{F}))$, which is a subsheaf of $\mathcal{F} \in \underline{\text{Ab}}_X$, is flasque when $\mathcal{F}$ is flasque. Here, $Z$ is a closed subset of $X$, and $\Gamma_{Z\cap U}(U,\mathcal{F})$ are the sections of $\mathcal{F}(U)$ that have support in $Z\cap U$.
Since it suffices to show surjectivity for $\Gamma_Z(X,\mathcal{F}) \to \Gamma_{Z\cap U}(U,\mathcal{F})$ for any $U \subset X$ open, we pick some $s \in \Gamma_{Z\cap U}(U,\mathcal{F})$ and try to extend it to $X$. Since the extended section must also have support in $Z$, we extend $s$ by zero outside $Z$. I'm sure I have to use the flasqueness of $\mathcal{F}$ together with sheaf properties, but I'm not really sure how to fix an open cover of $X$, given that we have information only about $U$ and $Z$. How do I proceed?
EDIT: I think I have the solution - Consider the open set $V \cup (X-Z)$ of $X$, if we have $t|_V = s$ and $t|_{X-Z} = 0$, then since they agree on the overlap, such $t$ must exist in $\mathcal{F}(V\cup(X-Z))$ by the sheaf property. Then since $\mathcal{F}$ is flasque, there is some $t^\prime \in \mathcal{F}(X)$ that agrees with $t$. Then $t^\prime$ is our desired section. I think this is correct, please let me know if something is missing.