In the proof of Robinson's "A course in the theory of groups", 3.1.8 (theorem due to Wilson) he mentions H-admissible subgroups, but without, afaik, defining that.
In this context, $H$ is a normal subgroup of $G$ of finite index, and we are assuming that $G$ satisfies min-$n$ but $H$ doesn't satisfy min-$n$ (for contradiction, since we are trying to prove $H$ does in fact satisfy min-$n$.) Here min-$n$ is the property of a group that it has no infinite, strictly descending chain of normal subgroups. So far so good.
Then without explanation he asserts "Now $H$ does not satisfy min$H$, the minimal condition on $H$-admissible subgroups," but I don't know the meaning of "$H$-admissible subgroups." The concept is then used in the long proof that I haven't yet understood - so any help would be appreciated.
For context, the next sentence is "By min-$n$ [on G, I suppose] if follows that $H$ contains a normal subgroup $K$ of $G$ which is minimal with respect to not satisfying min$H$." So I think we are going to take a strictly decreasing chain of normal subgroups of G, that are also subgroups of (and hence normal in) $H$, and that don't satisfy min$H$ and take the smallest of those (using min-$n$ on $G$.)
By the way it's asserted so casually, I assume "not min-$n$" should immediately imply "not min$H$" and so $H$-admissible on a subset of $H$ should immediately imply normality in $H$ (for then every $H$-admissible chain would also be a chain of normal subgroups of H.) But that it shouldn't be equivalent to normality in $H$ (because then, why would he bother to use it.)
So suppose $A\le G$, what does it mean for $A$ to be $H$-admissible?
I believe that if $A\le H$ then "$A$ is $H$-admissible" $\implies A\lhd H$ but not the converse.
I assume that $H$ is considered to act somehow on subgroups, analogous to $\Omega$-admissible. I considered two possible conditions (i) $HA=AH$ and (ii) $HA = A$. But if $A\le H$ the first is always true (since $H\lhd G$) and the second true only when $A=H$ so neither seems useful as a definition of $H$-admissible.
I'm stuck. I'm sure I'm missing something obvious, or likely made a logical mistake. Or both.