  In the category $\mathcal{P}_{d}$ of strict polynomial functors, the
morphisms between extension groups induced by the Frobenius twist are
injective. In \cite{Cuo14a}, the category $\mathcal{P}_{d}$ is proved to be a
full sub-category of the category $\mathcal{U}$ of unstable modules
\textit{via} Hai's functor. The Frobenius twist is extended to the category
$\mathcal{U}$ but remains mysterious there. This article aims to study the
Frobenius twist and its effects on the extension groups of unstable modules. We
compute explicitly several extension groups and show that in these cases, the
morphisms induced by the Frobenius twist are injective. These results are
obtained by constructing the minimal injective resolution of the free unstable
module $F(1)$.
