Consider a profinite group $S$ acting trivially on $\mathbb{F}_p$. Choose $\chi \neq 0\in H^1(S, \mathbb{F}_p)$ and set $T = \ker(\chi)$. Let $X$ be the $S$-Module of all functions $S/T \rightarrow \mathbb{F}_p$.
Show that there exists a canonical isomorphism $$ H^q(S,X) \cong H^q(T,\mathbb{F}_p). $$
The case $q=0$ is obvious, since $X^S$ consists of the constant functions and $\mathbb{F}_p^T = \mathbb{F}_p$ may be embedded into $X$ as the constant functions.
But how do I proceed from there? Dimension shifting doesn't seem to work here or am I missing something?
Thanks to i. m. soloveichik I was able to solve my question.
There exists an isomorphism
$$ \text{Ind}^T_S(\mathbb{F}_p) \cong \text{Map}(S/T, \mathbb{F}_p) = X $$ given by $$ x(\sigma) \mapsto y(\sigma T) = \sigma x (\sigma^{-1}) $$
By Shapiro's Lemma we get the isomorphism
$$ H^q(S, \: \text{Ind}^T_S(\mathbb{F}_p)) \cong H^q(T,\mathbb{F}_p). $$
A proof is found in Neukirch/Schmidt/Wingberg: 'Cohomology of Number Fields' p. 60.