Quotient-universal group property

Definition
A group property $$p$$ is termed quotient-universal if given any group $$G$$, there exists a group $$K$$ satisfying $$p$$ and a normal subgroup $$N$$ of $$K$$ such that $$G$$ is isomorphic to the quotient group $$K/N$$.

Note that the term quotient-universal is not standard, and some people use it in a narrower sense, for instance, only for finitely generated groups or only for finite groups.