# Is there a characterization of groups with the property $\exists N \unlhd G : \not \exists H \leq G \,\,\,s.t.\,\,\,H\cong G/N$?

A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$. Is there a characterization for groups for which this property fails?

If this question is too broad, I might ask if such a characterization exists for $p$-groups.

As @Ralph points out, we could restate this property as "groups in which at least one subgroup is not an endomorphism kernel." Evidently there is a bit of literature on groups for which the only endomorphism kernels are trivial or the whole group, but (outside of simple groups) this is a substantially stronger condition.

All finite abelian groups don't have the property, and non-abelian groups of order $\,p^3\,\,,\,\,p$ a prime have it. – DonAntonio Jan 15 at 23:31
If $G$ is a free group of rank $\ge 2$ then $G'$ has the required property. – Boris Novikov yesterday