Field automorphisms send roots into roots

Question

In the post Prove that a field automorphism sends a root into a root, the following problem is discussed:

If $E$ is an extension of $F$, $f(x) \in F[x]$ and $\phi$ is an automorphism of $E$ leaving every element of $F$ fixed, prove that $\phi$ must take a root of $f(x)$ lying in $E$ into a root of $f(x)$ in $E$.

I managed to solve it, using the argument described in the above post, but what I didn't see is where we used that $\phi$ is an auto-morphism. To me, it seems like it should work with any endo-morphism that fixes $F$. Is this true? If not, where were the hypothesis used in the proof of the above fact? And if it is indeed true, why is it?

Thanks in advance and sorry for the silly question!

Answer 1

Hint: if you had an endomorphism that wasn't an isomorphism what might its kernel be?