Let $E/K$ be an elliptic curve over a number field $K$ and let $E[n]$ denote the full $n$-torsion of $E$, for a positive integer $n$. With $End_K(E)$ we denote the endomorphisms of $E/K$ which are defined over $K$ and similarly we define $End_K(E[n])$ to be the endomorphisms of $E[n]$ which are defined over $K$. There is a natural restriction map $$\phi: End_K(E) \rightarrow End_K(E[n]).$$ I am interested in the case that the image of $\phi$ contains $Aut_K(E[n])$, the set of automorphisms of $E[n]$ which are defined over $K$. My question is the following:

Given $E/K$, is there always a lower bound $n_0$, such that for all $n>n_0$ we have that the image of $\phi$ contains $Aut_K(E[n])$?

Are there any differences between non-CM or CM curves?

Are there any differences between $\mathbb Q$ or a general number field $K$?

If the answer to the above question is yes, are there any results or conjectures concerning a global bound $n_0(K)$ which is valid for all elliptic curves over a fixed $K$?