  The Post Correspondence Problem is a classical decision problem about
equalisers of free monoid homomorphisms. We prove connections between several
variations of this classical problem, but in the setting of free groups and
free group homomorphisms. Among other results, and working under certain
injectivity assumptions, we prove that computing the rank of the equaliser of a
pair of free group homomorphisms can be applied to computing a basis of this
equaliser, and also to solve the "generalised" Post Correspondence Problem for
free groups.
