Let $V$ be a $\mathbb{Z}_2$-graded vector space (aka super vector space) and $L(V)$ be the free $\mathbb{Z}_2$-graded Lie algebra (aka super Lie algebra). The free super Lie algebra is also graded by the number of generators (the generators $V$ sit in degree 1, and the Lie bracket itself has degree zero in this sense). So it has a decomposition by degrees, $L(V) = \sum_{n=1}^\infty L^n(V)$. The homogeneous components $L^n(V)$ may be called (super) Lie powers of $V$ (by analogy with symmetric or alternating powers) and have an embedding $L^n(V) \subset V^{\otimes n}$, where the Lie (super)commutators are sent to tensor (super)commutators, $[a,b] = a\otimes b - (-)^{|a||b|} b \otimes a$.
Q: What is the $GL(V)$$\mathrm{GL}(V)$-representation theoretic description of $L^n(V)$?
When $V$ is purely even (has no odd component), the question is answered by Klyachko's theorem (as explained in the answers to MO187545). So basically, I'm asking: What is the super Lie algebra analog of Klyachko's theorem? And what is a reference that clearly states it?
