This is question from Silverman's Advanced topics in the arithmetic of elliptic curves, p. 106. Let $E_1$ and $E_2$ be elliptic curves. What does
isogeny $φ:E_1→E_2$ is determined by its kernel, at least up to an automorphism of $E_1$ and $E_2$
mean? Once $\text{ker}\, φ$ is given, what can we say about $φ$ and what does it have to do with $\text{Aut}(E_1)$ and $\text{Aut}(E_2)$ ? (To me, $\text{Aut}$ is nothing to do with $φ$).
Here's an analogue in group theory that is pretty arrow-theoretic (meaning it's about general nonsense with mappings that you see in many categories having quotients) and thus applies to modules, vector spaces, and so on.
Let $f \colon G_1 \to G_2$ and $f' \colon G_1 \to G_2$ be surjective group homomorphisms between the same groups with the same kernel $K$. Are $f$ and $f'$ related? They induce isomorphisms $\overline{f}, \overline{f'} \colon G_1/K \to G_2$. Then $F = \overline{f} \circ \overline{f'}^{-1} \colon G_2 \to G_2$ is an automorphism of $G_2$ and $F \circ \overline{f'} = \overline{f}$ as isomorphisms $G_1/K \to G_2$, and pulling back to $G_1$ we get $F \circ f' = f$ as homomorphisms $G_1 \to G_2$. Thus the homomorphisms $f$ and $f'$ with the same kernel are the "same" map up to an automorphism of $G_2$.