TITLE: Is there a characterization of groups in which at least one subgroup is not an endomorphism kernel?
QUESTION [15 upvotes]: This is a crosspost from MSE: 
Is there a characterization of groups with the property $\exists N \unlhd G : \not \exists H \leq G \;\; \text{s.t.} \;\; H\cong G/N$?

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?

REPLY [2 votes]: In short, I do not think there is a reasonable characterization of such groups, beyond restating the definition you gave. Here is why. First, let's give the property of $G$ that every quotient $G/N$ embeds in $G$ "self-universality" (SU). 
First, note that there are three classes of groups which have SU:

Simple groups (and groups obtained from them via semidirect products). 
Higman-Neumann-Neumann type universal groups: They proved existence of 2-generated groups $G$ which contain every countable subgroup. In particular, such $G$ contains every quotient of $G$. 
Note that in (1), the group has too few quotients, while in (2) it has too many subgroups. One can have infinite f.g. groups which have both too few quotients and too few subgroups, like Tarski monsters (every nontrivial proper subgroup is isomorphic to $Z_p$).   

Given the diverse reason for these groups to have SU, there does not seem to be any "unifying reason" for the SU property. 
Note, however, that, at least among finitely-generated groups, all the examples 1-3 are mostly "artificial"; e.g.: most "naturally occurring" f.g. groups tend not to be simple (or universal in HNN sense, or be "monsters"). Among "natural" f.g. groups, the ones which do not have SU tend to dominate:
a. F.g. (infinite) matrix groups (over the real numbers) do not have SU, by a combination of Malcev's theorem and Selberg's lemma: If $G$ is such a group, it has finite quotients of arbitrarily high orders (Malcev), but $G$ contains only  finitely many isomorphism classes of finite subgroups (Selberg). 
b. Infinite hyperbolic groups do not have SU: For virtually cyclic groups this follows from (a), while for nonelementary groups it follows from Delzant and Olshansky who proved SQ universality for such groups. In particural, such $G$ always has a quotient containing $Z^2$ and, hence, cannot embed in $G$.