Let $K$ be a local field with perfect residue field of characteristic $p>0$ prime, and denote by $G_K$ its absolute Galois group. Let $V$ be a mod-$p$ representation of $G_K$, i.e. a finite dimension $\mathbb{F}_p$-vector space together with a linear and continuous action of $G_K$. Now we construct the following module associated to $V$ by $Ind_{G_K}(V):=Fct_{cont}(G_K, V)$, the set of all continuous maps from $G_K$ to $V$. Endow $Ind_{G_K}V$ with the discrete topology and the action of $G_K$:
\begin{align*}
G_K \times Ind_{G_K}(V) &\to Ind_{G_K}(V) \\
g\cdot \eta &\mapsto [x\mapsto \eta(x\cdot g)]
\end{align*}
$V$ then canonically injects into $Ind_{G_K}(V)$ by sending $v\mapsto \eta_v$, where $\eta_v(g)=g(v)$ for any $g\in G_K$.
(1) $Ind_{G_K}(V)$ should be of infinite dimensional as a $\mathbb{F}_p$-module. What is a basis of it?
(2) The Galois action on $Ind_{G_K}(V)$ seems to be transitive: solving $g\cdot \eta=h$ gives that $\eta(-)=h(-\cdot g^{-1})$, which should again lie in $Ind_{G_K}(V)$. In other words, $g$ is surjective on $Ind_{G_K}(V)$. What can you say about the image of $g-1$ on $Ind_{G_K}(V)$? Can you have a simple characterization of the image, and for which group element $g$ the image of $g-1$ is big?
(3) Suppose $g^{p^n}\to 1, n\to \infty$, can you say more on the image of $g-1$ over $Ind_{G_K}(V)$now?