Let $q$ be a prime power and let $m$ be a positive integer coprime to $q$. Further, let $E_1(\mathbb F_q)$ and $E_2(\mathbb F_q)$ be two isogenous supersingular elliptic curves over a finite field $\mathbb F_q$ such that

$$E_i(\mathbb F_q)[m] \cong \mathbb Z/m\mathbb Z \times \mathbb Z/m\mathbb Z$$

for $i = 1, 2$. Suppose that an isogeny $\phi \colon E_1 \rightarrow E_2$ is separable and has degree coprime to $m$. So in particular $\phi$ is a one-to-one and onto map between $E_1[m]$ and $E_2[m]$.

Now suppose that you are given a pair of generators $P,Q$ of $E_1[m]$ and a pair of generators $R,S$ of $E_2[m]$. In this setting, I have three questions.

1. For which matrices

$$
\left(
\begin{matrix}
a & b\\
c & d
\end{matrix}
\right)
\in \operatorname{GL}_2(\mathbb Z)
$$

does there exist an endomorphism $\sigma \colon E_1 \rightarrow E_1$ such that $\sigma(P) = aP+bQ$ and $\sigma(Q)=cP+dQ$?

2. How hard is it to answer the following question: *does there exist an isogeny $\phi \colon E_1 \rightarrow E_2$ such that $\phi(P)=R$ and $\phi(Q) = S$?* Of course in general it is difficult to compute $\phi$, but perhaps answering this question of existence is easier?

3. Given a matrix

$$
\left(
\begin{matrix}
a & b\\
c & d
\end{matrix}
\right)
\in \operatorname{GL}_2(\mathbb Z)
$$

and a fact that $\phi(P)=R$ and $\phi(Q)=S$ for some isogney $\phi$, does there exist an isogeny $\phi'$ such that $\phi'(P) = aR+bS$ and $\phi'(Q)=cR+bS$? For which matrices is this possible?