We present a systematic study of the orbifolds of the rank $n$ symplectic fermion algebra $\mathcal{ A}(n)$, which has full automorphism group $Sp(2n)$. First, we show that $\mathcal{ A}(n)^{ Sp(2n)}$ and $\mathcal{ A}(n)^{ GL(n)}$ are $\mathcal{ W}$-algebras of type $\mathcal{ W}(2,4,\dots, 2n)$ and $\mathcal{ W}(2,3,\dots, 2n+1)$, respectively. Using these results, we find minimal strong finite generating sets for $\mathcal{ A}(mn)^{ Sp(2n)}$ and $\mathcal{ A}(mn)^{ GL(n)}$ for all $m,n\geq 1$. We compute the characters of the irreducible representations of $\mathcal{ A}(mn)^{ Sp(2n)\times SO(m)}$ and $\mathcal{ A}(mn)^{ GL(n)\times GL(m)}$ appearing inside $\mathcal{ A}(mn)$, and we express these characters using partial theta functions. Finally, we give a complete solution to the Hilbert problem for $\mathcal{ A}(n)$; we show that for any reductive group $G$ of automorphisms, $\mathcal{ A}(n)^G$ is strongly finitely generated.
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