We show that the affine vertex superalgebra $V^k(\mathfrak{ osp}_{ 1|2n})$ at generic level $k$ embeds in the equivariant $\mathcal W$-algebra of $\mathfrak{ sp}_{ 2n}$ times $4n$ free fermions. This has two corollaries: (1) it provides a new proof that for generic $k$, the coset $\text{ Com}(V^k(\mathfrak{ sp}_{ 2n}), V^k(\mathfrak{ osp}_{ 1|2n}))$ is isomorphic to $\mathcal W^\ell(\mathfrak{ sp}_{ 2n})$ for $\ell = -(n+1) + \frac{ k+n+1}{ 2k+2n+1}$, and (2) we obtain the decomposition of ordinary $V^k(\mathfrak{ osp}_{ 1|2n})$-modules into $V^k(\mathfrak{ sp}_{ 2n}) \otimes \mathcal W^\ell(\mathfrak{ sp}_{ 2n})$-modules. Next, if $k$ is an admissible level and $\ell$ is a non-degenerate admissible level for $\mathfrak{ sp}_{ 2n}$, we show that the simple algebra $L_k(\mathfrak{ osp}_{ 1|2n})$ is an extension of the simple subalgebra $L_k(\mathfrak{ sp}_{ 2n}) \otimes { \mathcal W}_{ \ell}(\mathfrak{ sp}_{ 2n})$. Using the theory of vertex superalgebra extensions, we prove that the category of ordinary $L_k(\mathfrak{ osp}_{ 1|2n})$-modules is a semisimple, rigid vertex tensor supercategory with only finitely many inequivalent simple objects. It is equivalent to a certain subcategory of $\mathcal W_\ell(\mathfrak{ sp}_{ 2n})$-modules. A similar result also holds for the category of Ramond twisted modules. Due to a recent theorem of Robert McRae, we get as a corollary that categories of ordinary $L_k(\mathfrak{ sp}_{ 2n})$-modules are rigid.
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