I have been trying to construct the Weil Group for an arbitrary class formation $(G,C)$ where $G$ is profinite. We assume that $C$ has trivial universal norms (ie. for any open subgroup $U \subset G$ have $\bigcap_{V \subset U} N_{U/V}(C^V) $ is trivial) and the quotients $C^V/C^U$ are all compact . ( although I don't this we need these conditions for my question below, I'm following Chapter 2 Section 1 of https://link.springer.com/book/10.1007/978-3-540-37889-1.)
Where I'm getting confused is the following.
Take opens $W \subset V \subset U$ where $W,V$ are normal in $U$. We have each of $C^W$ is a class module for $U/W$ with a fundamental class $u_{U/W} \in H^2(U/W, C^W)$, and similarly for $U/V$. Then we can take the associated group extensions of these cohomology classes to get groups $\mathcal{W}(U/W)$ and $\mathcal{W}(U/V)$.
Now, my aim is to construct an inverse system of these groups for any fixed $U$, so we need transition maps $\phi_{V/W} : \mathcal{W}(U/W) \to \mathcal{W}(U/V)$. The natural way one may look for these is to seek a homomorphism such that the following diagram commutes.
Link to diagram, as I can't add it inline
Its a fact about group extensions that such a vertical map $F$ exists, if and only if the maps $N_{V/W}$ and $\pi$ are compatible in the following sense.
(1) $N_{V/W}$ is a $U/W$ module map, where we endow $C^V$ with $U/W$ module structure via $\pi$.
(2) We have $N_{V/W *}(u_{U/W})= \pi^* (u_{U/V})$ where the maps are induced by the compatible maps $(U/W \xrightarrow{id}U/W, C^W \xrightarrow{N_{V/W}} C^V)$ and $(U/W \xrightarrow{\pi} U/V, C^V \xrightarrow{id} C^V)$.
However, I can't seem to prove this second condition?
