= Solution
{c}
For $X\in\operatorname{SO}_3(\mathbb R)$, the defining identity $X^TX=I$ gives
$$
\|Xv\|^2=v^TX^TXv=v^Tv=\|v\|^2.
$$
Thus an <orthogonal transformation> preserves the <Euclidean norm> and sends the <unit sphere> to itself. Restricting the given <group action> to this invariant set preserves both action axioms.
If $XN=N$, the first column of $X$ is $N=e_1$. The other columns are <unit vectors> perpendicular to it, so the first row is also $(1,0,0)$. Hence $X=\operatorname{diag}(1,B)$ with $B^TB=I_2$ and $\det B=1$. Conversely, every such <matrix> lies in $G$ and fixes $N$. Since every element of the <special orthogonal group in two dimensions> has the planar <rotation matrix> form, the <stabilizer subgroup> is
$$
\boxed{H=\left\{R_\theta=
\begin{pmatrix}
1&0&0\\
0&\cos\theta&-\sin\theta\\
0&\sin\theta&\cos\theta
\end{pmatrix}:\theta\in\mathbb R\right\}\cong\operatorname{SO}_2(\mathbb R).}
$$
Geometrically these are exactly the <rotations in three dimensions> about the first coordinate axis.
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