In Embedding theorems for groups, (J. London Math. Soc. 34 1959 465–479.) Neumann and Neumann (NB: this is not the Higman-Neumann-Neumann paper of the same name) make the following definition.
Definition: A subgroup $H$ of a group $G$ is an E-subgroup of $G$ if for every normal subgroup $N\triangleleft H$, there is a normal subgroup $S\triangleleft G$ such that $S\cap H=N$. Equivalently, for every normal subgroup $R\triangleleft H$, the normal closure $R^G$ of $R$ in $G$ is such that $R^G\cap H=R$.
This comes up again in The SQ-universality of hyperbolic groups, by Olshanskii, wherein he simply refers to this situation as the normal structure of $H$ being a restriction of the normal structure of $G$.
Presumably this notion has come up elsewhere, although I don't know of other places where it is explicitly named. It's a natural notion to deal with when working to show a group is SQ-universal, for example. Lately I've been using it in my research, and I would like to know if there is an agreed-upon name for this type of subgroup. As far as I can tell, "E-subgroup" is not widely known. Is there something that is?