[I'm self-studying group theory after many, many years away, using Derek Robinson's book.]
Robinson in Ex 2.2.8 says:
Let G be a finitely presented group and let N be a normal subgroup which is finitely generated as a G-operator group. Prove that G/N is finitely presented.
I think I understand all the words like "G-operator" and "finitely generated," but I don't know what they mean combined (as emphasised in this sentence.)
[Note: I'm not asking for a solution to the problem (yet!) but just to know what it means.]
I don't think Dietrich's guess in the comments is correct (the word "as" is relevant). My guess is that "finitely generated as a $G$-operator group" means that there exists a finite subset $S$ of $N$ such that every element of $N$ can be obtained from this subset by repeated multiplication and the action of $G$ (by conjugation). Equivalently, $N$ is generated (as a group) by $\cup_{g \in G} gSg^{-1}$.
Note that the exercise is much easier if you just assume that $N$ is finitely generated but then there's no need to mention the concept of a $G$-operator group at all.