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Finite subgroup of $SO$\mathrm{SO}(4)$ which acts freely on $S^3$$\mathbb{S}^3$

Let $\Gamma$ be a finite subgroup of $SO(4)$$\mathrm{SO}(4)$ acting freely on $S^3$$\mathbb{S}^3$. It is known that all such $\Gamma$ can be classified.

Is there any characterization of $\Gamma$ such that $\Gamma$ is conjugate in $O(4)$$\mathrm{O}(4)$ to a subgroup of $U(2)$$\mathrm{U}(2)$?

Finite subgroup of $SO(4)$ which acts freely on $S^3$

Let $\Gamma$ be a finite subgroup of $SO(4)$ acting freely on $S^3$. It is known that all such $\Gamma$ can be classified.

Is there any characterization of $\Gamma$ such that $\Gamma$ is conjugate in $O(4)$ to a subgroup of $U(2)$?

Finite subgroup of $\mathrm{SO}(4)$ which acts freely on $\mathbb{S}^3$

Let $\Gamma$ be a finite subgroup of $\mathrm{SO}(4)$ acting freely on $\mathbb{S}^3$. It is known that all such $\Gamma$ can be classified.

Is there any characterization of $\Gamma$ such that $\Gamma$ is conjugate in $\mathrm{O}(4)$ to a subgroup of $\mathrm{U}(2)$?

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Adterram
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Finite subgroup of $SO(4)$ which acts freely on $S^3$

Let $\Gamma$ be a finite subgroup of $SO(4)$ acting freely on $S^3$. It is known that all such $\Gamma$ can be classified.

Is there any characterization of $\Gamma$ such that $\Gamma$ is conjugate in $O(4)$ to a subgroup of $U(2)$?