  To any Frobenius superalgebra $A$ we associate an oriented Frobenius Brauer
category and an affine oriented Frobenius Brauer categeory. We define natural
actions of these categories on categories of supermodules for general linear
Lie superalgebras $\mathfrak{gl}_{m|n}(A)$ with entries in $A$. These actions
generalize those on module categories for general linear Lie superalgebras and
queer Lie superalgebras, which correspond to the cases where $A$ is the ground
field and the two-dimensional Clifford algebra, respectively.
