Let $f:G\to H$ be a homomorphism of finitely presented groups with decidable word problems.

Assume you are given explicit finite presentations for both $G$ and $H$ and you are given the words to which $f$ sends the generators of $G$.

Question.Is it possible that no algorithm exists deciding, given $w\in H$, whether $f^{-1}(w)$ is empty or not?

P.S. Having explicit finite presentations is important https://arxiv.org/abs/1003.5117

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