Hopficity of profinite completions of abelian groups

We determine exactly when the profinite completion of an arbitrary abelian group is topologically Hopfian. For an abelian group $A$, we prove that $$\widehat{A}\text{ is topologically Hopfian}⟺A\mathrm{/}pA\text{ is finite for every prime }p\mathrm{.}$$ As a byproduct, we answer Problem 6.30 of the Kourovka Notebook in the negative: for pairwise distinct odd primes $q_i$, the group ${⨁}_{i\ge 1}\mathbb{Z}[1\mathrm{/}{q}_i]$ is residually finite and Hopfian, whereas its profinite completion is not topologically Hopfian.