  We define an action of the degenerate two boundary braid algebra
$\mathcal{G}_d$ on the $\mathbb{C}$-vector space $M\otimes N\otimes V^{\otimes
d}$, where $M$ and $N$ are arbitrary modules for the general linear Lie
superalgebra $\mathfrak{gl}(n|m)$, and $V$ is the natural representation. When
$M$ and $N$ are parametrized by rectangular hook Young diagrams, this action
factors through a quotient $\mathcal{H}^{\operatorname{ext}}_d$. The
irreducible summands of $M\otimes N\otimes V^{\otimes d}$ for the centralizer
of $\mathfrak{gl}(n|m)$, remain irreducible once regarded as modules for this
quotient $\mathcal{H}^{\operatorname{ext}}_d$.
