For $n$ large enough the Galois group of $F_n/F_{n-1}$ identifies with the kernel $G$ of $\operatorname{GL}_2(\mathbb{Z}/p^{n+1}\mathbb{Z}) \to \operatorname{GL}_2(\mathbb{Z}/p^{n}\mathbb{Z})$. Let $P$ and $Q$ be a basis of $E_{p^{n+1}}$. The norm of $P$ is written as a sum over $a$, $b$, $c$, $d$ modulo $p$ as
\begin{align*} N_G(P) &= \sum_{a,b,c,d} \begin{pmatrix} 1+ap^n & bp^n \\ cp^n & 1+dp^n\end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix}\\
&= \sum_{a,b,c,d}\bigl( (1+ap^n)P+cp^nQ\bigr)\\
&= p^4P+p^n(\sum_{a=0}^{p-1} a)\, P + p^n (\sum_{c=0}^{p-1} c)\, Q \\
&= p^4 P\end{align*}
and similar for $Q$ so it is multiplication by $p^4$ on all of $E_{p^{n+1}}$.