Background: Let $G$ be a profinite group, for $M \leqslant N$ open normal subgroups we have the projection map $p_{M,N} \colon G/M \to G/N$ which induces a transfer map on rational group rings $$ p_{M,N}^\# \colon \mathbb{Q}[G/N] \to \mathbb{Q}[G/M], \quad gN \mapsto \sum_{p_{M,N}(xM)=gN } xM $$ which we can describe as summing over the pre-image of $gN$ (or the sum of a transversal of $M$ in $N$), which is finite as $M$ and $N$ are open in $G$.
Using these maps we can take the direct limit over the open normal subgroups $N$ to construct a discrete $G$-module. $$\mathbb{Q}\langle G \rangle= \underset{N \trianglelefteq G}{\text{colimit}} \ \mathbb{Q}[G/N]$$
Question: Has anyone encountered this discrete $G$-module $\mathbb{Q}\langle G \rangle$ before?
I'm also interested in the more general case $\underset{N \trianglelefteq G}{\text{colimit}} \ [G/NK]$ for $K$ a closed subgroup of $G$.
Sub-question: If we replace $\mathbb{Q}$ by an ind-finite ring $R$, then this looks like the dual of the completed group ring over the dual of $R$. Is this correct? Is this dual used anywhere?
