2 specified ring isomorphisms

(This is inspired by the answer to my earlier question.)

Does there exist

a field $F$ $\:$ and $\:$ two ideals $I$ and $J$ of $F[x]$ $\:$ and $\:$ an a ring isomorphism $\: \phi : F[x]/I \to F[x]/J$

such that when $\: q_i : F[x] \to F[x]/I \:$ and $\: q_j : F[x] \to F[x]/J \:$ are the quotient maps,
there does not exist an endomorphism $\: \psi : F[x] \to F[x] \:$ such that $\;\; \phi \circ q_i \: = \: q_j \circ \psi \;\;$?

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# Are there ever exotic isomorphisms between quotients of F[x]?

(This is inspired by the answer to my earlier question.)

Does there exist

a field $F$ $\:$ and $\:$ two ideals $I$ and $J$ of $F[x]$ $\:$ and $\:$ an isomorphism $\: \phi : F[x]/I \to F[x]/J$

such that when $\: q_i : F[x] \to F[x]/I \:$ and $\: q_j : F[x] \to F[x]/J \:$ are the quotient maps,
there does not exist an endomorphism $\: \psi : F[x] \to F[x] \:$ such that $\;\; \phi \circ q_i \: = \: q_j \circ \psi \;\;$?