What does 'Isogeny is determined by its kernel up to automorphisms' mean?

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 $$φ$$).

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 $$G_2$$.