In what follows, all groups are assumed to be finite.

Recall that if $K \leq H \leq G$ are groups, $K$ is said to be a *weakly closed* subgroup of $H$ in $G$ if, for all $g \in G$, $g^{-1}Kg \leq H$ implies that $g^{-1}Kg = K$.

The subgroup property I am interested in is this one: what kind of subgroup of $H$ is $K$ when, for all groups $G$ with $H \leq G$, $K$ is a weakly closed subgroup of $H$ in $G$?

It is trivial to verify that $1$ and $H$ are subgroups of $H$ with this property. (Here, already, the finiteness condition is important: if $H$ is infinite, one could have $g^{-1}Hg < H$ instead of having $g^{-1}Hg = H$. So I won't go there.)

It is clearly necessary that $K$ be a characteristic subgroup of $H$ for this unnamed property to hold:

If $H$ is finite, so is its holomorph $Hol(H) = H \rtimes Aut(H)$. So let $G = Hol(H)$. Then, regarding $Aut(H)$ as a subgroup of $G$, let $\alpha \in Aut(H)$. Then $\alpha^{-1}K\alpha \leq H$ because $\alpha$ is an automorphism of $H$. Since $K$ is, by assumption, a weakly closed subgroup of $H$ in $G$, this means $\alpha^{-1}K\alpha = K$. Since $\alpha \in Aut(H)$ was arbitrary, it follows that $K$ is a characteristic subgroup of $H$.

On the other hand, merely being characteristic is not sufficient:

Let $G = <(1243675), (4657)(23)>$, so that $G \cong GL_{3}(2)$. Let $H = <(4657)(23), (67)(23)>$ and $K = <(45)(67)>$. Then $K = Z(H)$, so $K$ is a characteristic subgroup of $H$. It is also true that $H \leq G$, and that $g = (13)(57) \in G$.

But then $g^{-1}Kg = <(47)(65)> \leq H$, but $<(47)(65)> \neq < (45)(67) >$.

I have heard of fully invariant subgroups, but this also shows that a fully invariant subgroup need not have this property: in the example above, $Z(H)$ is a fully invariant subgroup of $H$ because $Z(H)$ is the subgroup generated by the squares in $H$.

What I can say is that if a subgroup $K$ of $H$ is the only subgroup of $H$ with elements having the orders they do, then $K$ is this kind of subgroup of $H$. (So, in the above example, $<(4657)(23)>$ is the unique cyclic subgroup of order 4 in $H$ and it is this kind of subgroup of $H$, unlike $Z(H)$.)

Does this kind of subgroup have a name? What other sets of conditions on a characteristic subgroup imply this stronger property?