Decide the existence of a string homomorphism
It's discussed in one of the very first papers about strings and complexity, namely, Dana Angluin, Finding patterns common to a set of strings, J. Comput. System Sci.
21 (1980), 46-62.  Look at Theorem 3.6.  The problem is NP-complete.

It's also in A. Ehrenfeucht, G. Rozenberg, Finding a homomorphism between two words is NP-complete, Inform. Process. Lett. 9 (1979) 86–88.