Quotient-universal group property

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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.