project limit on $n$- simplical complex which is principal homogeneous with respect to an action

Question

The setting:

Let G be compact locally $\Bbb{Q}_p$ analytic group. We fix a countable basis of open normal subgroups $G\supset G_1\supset ...G_r\supset...$ We suppose that we are given a system of continuous maps of topological spaces ....$->X_r->...X_1->X_0$. we assume that theses spaces are equipped with a right action of $G$ such that

1. The maps in the sequence are $G$ equivariant.

2. The action of $G_r$ on $X_r$ is trivial

3. The map $X_p->X_q$ is Galois covering map with deck transformations provided by the natural action of $G_q/G_p$ on $X_r$.

Also assume that $X_0$ is a finite simplical complex. Let $T(X_0)$ be the choice of finite triangulation of $X_0$. let $T(X_r)$ denote the pullback of $T(X_0)$ under the finite covering map $X_r->X_0$. Let $\delta \in T_n(X_0) $(which is by definition the collection of n dimensional simplices occuring in the triangulation) then we let $T_n(X_r)_{/\delta}$ denote the set of simplices in $T_n(X_r) $ lying over $\delta$.

How to show the following:

1. that $T_n(X_r)_{/\delta}$ is then a principla homogeneous $G/G_r$ set.
2. The projective limit of $T_n(X_r)_{/\delta}$ along $r$ is a profinite set which is principal homogeneous with respect to $G$ action.