Let $K,~Q$ be two groups, $K$ is abelian and $K$ is also a $\Bbb Z[Q]$-module. Then a group extension of $K$ by $Q$: $0\to K\to G\to Q\to 1$ realizes the operator if the scalar multiplication $Ck=\ell(C)\,k\,\ell(C)^{-1}$ for every $C\in Q,~k\in K$. ($\ell:Q\to G$ is a lifting.)
I know this definition, and I even can memorize it. However, I feel painful whenever see a theorem require it as one of its hypothesis. What does "realizes the operator" really mean? What concept does such definition want to catch? If the scalar multiplication of the module is the same as we use a lifting to produce a conjugation of $K$, then so what?
I've never seen it expressed like that, I would say something like "induces the $Q$-module structure on $K$" or something similar.
"realizes" corresponds to my "induces" and "the operator" corresponds to "the $Q$-module structure on $K$".
Indeed, from the fact that $K$ is a $\mathbb{Z}[Q]$-module, you have one action of $Q$ on $K$ (by definition); and from the extension $0\to K\to G\to Q\to 1$ you have another one, which is given by conjugation (and, $K$ being abelian, this is well defined): that the extension "realizes the operator" is to say that those two actions of $Q$ on $K$ coïncide.
In particular, when you study group cohomology for instance, $H^2(Q,K)$ will classify the extensions of $Q$ by $K$ that "realize the operator" (or as I would put it, the extensions that induce the correct action on $K$)