Proper symmetry group/Finite/Three classes of semiaxes/Operation injective/Exercise

Let $G\subseteq {\mathrm{SO}}_3\left(ℝ\right)$ be a finite subgroup of the group of proper linear isometries in ${ℝ}^3$ with three classes of semiaxes, and let $K$ denote one of them. Show that the group homomorphism

$G⟶\mathrm{Perm}(K),g⟼{\sigma }_g:H\mapsto g(H),$

is injective. Show that this is not true when there are only to classes of semiaxes.

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