As far as I know, for any group $G$ there exists an acyclic group $H$ such that $G$ is a subgroup of $H$.
I am wondering about the dual situation. Is any group $A$ a quotient of an acyclic group $B$ or more simply, given a group $A$ does it exist an acyclic group $B$ and a surjective homomorphism $B\rightarrow A$ ?



Any group is a quotient of an acyclic group?
Acyclic groups must in particular have trivial abelianization, so all of their quotients must be perfect.


This is the only obstruction; A.J. Berrick shows in [The acyclic group dichotomy](https://arxiv.org/abs/1006.4009) (which I just found by googling!) that every perfect group is a quotient of an acyclic group of cohomological dimension $2$ (Proposition 2.3).


