What is the name of the quotient of the Stiefel manifold of $k$-frames by the symmetric group of $k$ letters?

Let $V_k(\mathbb{R}^n)$ be the Stiefel manifold of $k$-frames in $\mathbb{R}^n$. The symmetric group of $k$ letters $\Sigma_k$ acts freely by permuting vectors in $k$-frames.

Does the quotient $V_k(\mathbb{R}^n)/\Sigma_k$ have a name?

This space seems to play an important role in the problem of $k$-regular embeddings. For example, seethis paper. But I've never heard of its name.

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