This is a crosspost from MSE:
A common mistake among 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 in which this assumption fails?
If this question is too broad, I might ask if such a characterization exists for $p$-groups.
A peripheral question is how likely it is for a finite group to have this property, which I would formulate in the following way: if $g(n)$ denotes the fraction of isomorphism classes of finite groups of order $≤n$ for which the property holds, what happens to $g(n)$ as $n\rightarrow \infty$? (Here is what $g$ looks like for small values.) If we again restrict to $p$-groups, can we say anything about the fraction of isomorphism classes of order $p^k$ with the property, e.g. a bound in terms of the exponent?
EDIT I believe Derek Holt's answer satisfies the peripheral question. The primary question is still open - can we say anything to characterize these groups?