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.
Update: I crossposted to MO and received an answer to the (now omitted) peripheral question about probability. The characterization is still unanswered.