= Solution
Take $X=\mathbb R$, $U=(0,1)$, and let $\mathcal F$ be the <skyscraper sheaf> with value a nonzero abelian group $A$ at $p=3/4$. Cover $V=(-1,1)$ by $V_1=(-1,2/3)$ and $V_2=(1/2,1)$. The presheaf gives $(j^p\mathcal F)(V)=0=(j^p\mathcal F)(V_1)$ and $(j^p\mathcal F)(V_2)=A$. A nonzero section on $V_2$ and the zero section on $V_1$ agree on the overlap, whose skyscraper sections vanish, but cannot be glued on $V$. Thus $j^p\mathcal F$ need not be a sheaf.
<Sheafification> preserves <stalk of a sheaf>[stalks]. If $P\in U$, neighborhoods contained in $U$ are cofinal, so $(j^p\mathcal F)_P=\mathcal F_P$. If $P\notin U$, no neighborhood of $P$ lies in $U$, so every term defining the presheaf stalk is zero. Therefore
$$
(j_!\mathcal F)_P\cong
\begin{cases}
\mathcal F_P,&P\in U,\\
0,&P\notin U.
\end{cases}
$$
Solved by gpt-5.6-sol high.
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