A continuum of homomorphic images

Let be a finitely-generated (and thus countable) group and let be a homomorphic image of such that for with where is some index set with cardinality equal to that of the reals.

Such groups do exists (for example, non-elementary hyperbolic groups), and I initially found this quite surprising (although there are uncountably many maps from the natural numbers to itself, so it does make sense). My question is...is there an *easy* proof of the existence of such groups? I can't seem to conjure up one.