Is there a name for a subgroup $H < G$ which is not necessarily normal, but such that, for any $f : G \rightarrow Y$ where $Y$ belongs to a particular category (e.g. finite groups, linear groups, etc.), we have $f(H) \trianglelefteq f(G) \qquad$ ?
We could imagine generalizing this concept almost indefinitely: Fix a category $\mathcal{C}$, a subcategory $\mathcal{C}_0$, and a property $P$ enjoyed by certain objects of $\mathcal{C}_0$. We could then consider objects $X$ of $\mathcal{C}_0$ such that, for any object $Y$ of $\mathcal{C}_0$ and epimorphism $f : X \rightarrow Y$, the codomain $Y$ enjoys said property.
If I haven't made a mistake, the situation in the first paragraph could then be obtained by taking $\mathcal{C}$ to be the category of pairs $(H, G)$ with $G$ a group and $H \leq G$ a subgroup, and filling in the remaining conditions in an appropriate way.
Similarly, if we take $\mathcal{C}$ to be the category of topological spaces, $\mathcal{C}_0$ to be the subspaces of $\mathbb{R}$, and $P$ to be boundedness, then we'd get the notion of pseudocompactness.
[ I have no doubt that I'm playing disgustingly fast-and-loose with categorical notions here -- corrections are very welcome. ]
My question: is there (i) a word for the notion in the first paragraph, or (ii) a general nomenclature (along the lines of "pro-X" or "residually-X" or ...) that lets us describe the situation in the second paragraph?