Sorry for my bad English.
Let $p$ be a prime number, $N'>N$ be coprime positive integer and $f:E_1\to E_2$ be a degree $N$ separable isogeny map between elliptic curves (if necessary supersingular).
When given basis $P,Q\in E_1[N']$ and the image $f(P), f(Q)$, how many $N$-isogenies $g:E_1\to E_2$ are there with $g(P)=f(P)$ and $g(Q)=f(Q)$?
By assumption there is at least example $f=g$, but when $g$ is uniquely determined?