A continuum of homomorphic images

Let $\displaystyle G$ be a finitely-generated (and thus countable) group and let $\displaystyle H_i$ be a homomorphic image of $\displaystyle G$ such that $\displaystyle H_i \not\cong H_j$ for $\displaystyle i \neq j$ with $\displaystyle i, j \in I$ where $\displaystyle I$ 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.