= Solution
Put $N=S^d/X$ and let $p=q(X)$. The restriction
$$
q:S^d\setminus X\longrightarrow N\setminus\{p\}
$$
is a homeomorphism. At every point of $N\setminus\{p\}$, transport the local orientation of $S^d$ through this homeomorphism. The localization of the global class $q_*[S^d]$ is therefore a generator at one, and hence every, point of that connected open set. The localizations of a global homology class form a section of the orientation local system; because the manifold $N$ is connected, this generator extends across $p$. Thus $q_*[S^d]$ is a fundamental class for an orientation of $N$.
Choose $[N]=q_*[S^d]$. By the <degree of a map between oriented manifolds>,
$$
q_*[S^d]=1\,[N],
$$
so $\boxed{\deg q=1}$.
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