Let $E_1,~E_2$ be two elliptic curve on the $p$-adic field $K \supseteq \mathbb{Q}_p$.
Consider the $p$-power torsion points and adjoin them with $K$.
Denote $E_i[p^n], ~i=1,2$ to be the set of $p^n$-torsion points of $E_i,~ i=1,2$, where $n$ some positive integers.
Denote the extensions $K_n=K(E_1[p^n])$ and $K_n'=K(E_2[p^n])$.
Case I: Consider the corresponding Galois extensions $G_1=\text{Gal}(K_n/K)$ and $G_2=\text{Gal}(K_n'/K)$.
Case II More generally, If further assume $\Lambda=\bigcup_{p,n} E_1[p^n]$ and $\Lambda'=\bigcup_{p,n} E_2[p^n]$, then consider $G_1=\text{Gal}(K(\Lambda)/K)$ and $G_2=\text{Gal}(K(\Lambda')/K)$.
My question-
For both above two cases,
If $n \to \infty$, does the Galois groups $G_1$ and $G_2$ are isomorphic under
some suitable assumptions?
Does $G_1$ and $G_2$ are map-able by a good map under
some suitable assumptionswhen $n \to \infty$?
Any intuitive idea, discussions are appreciated. Thanks
