Consider the following problem:
Given two strings x,y, decide whether there exists a string homomorphism f such that f(x)=y.
It is easy to show that this problem is in $NP$. Are there other things we can say about this problem? e.g. Is it in $coNP$, or even $P$?
This problem seems very natural, so I am not surprised if it has been studied thoroughly. However I could not find this problem in literature.