I've encountered in this paper (Remark 1. p. 7) the following terminology:

• A normal subgroup $N$ of $G$ is observable if every $N$-representation can be embedded $N$-equivariantly in the restriction of a $G$-representation.

Unfortunately, when I've tried to google what is known about groups with this property the literature only seems to talk about algebraic groups. Is there another term which is normally used for this property in the context of discrete groups?

A second question: are there any immediately known classes of subgroups which are always observable?

I guess finite index subgroups have this property, but I don't know any others.

I've encountered in Bass, Lubotzky, Magid, and Mozes - The proalgebraic completion of rigid groups (Remark 1. p. 7) the following terminology:

• A normal subgroup $N$ of $G$ is observable if every $N$-representation can be embedded $N$-equivariantly in the restriction of a $G$-representation.

Unfortunately, when I've tried to google what is known about groups with this property the literature only seems to talk about algebraic groups. Is there another term which is normally used for this property in the context of discrete groups?

A second question: are there any immediately known classes of subgroups which are always observable?

I guess finite index subgroups have this property, but I don't know any others.