Proper symmetry group/Finite/Classes/Group homomorphism/Kernel/Exercise

Let ${}G\subseteq \operatorname {SO} _{3}\!{\left(\mathbb {R} \right)}$ be a finite subgroup of the group of proper linear isometries in ${}\mathbb {R} ^{3}$ with a fixed class of semiaxes ${}K$. Determine the kernel of the group homomorphism

$G\longrightarrow \operatorname {Perm} \,(K),g\longmapsto \sigma _{g}:H\mapsto g(H).$

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