  We associate a diagrammatic monoidal category
$\mathcal{H}\textit{eis}_k(A;z,t)$, which we call the quantum Frobenius
Heisenberg category, to a symmetric Frobenius superalgebra $A$, a central
charge $k \in \mathbb{Z}$, and invertible parameters $z,t$ in some ground ring.
When $A$ is trivial, i.e. it equals the ground ring, these categories recover
the quantum Heisenberg categories introduced in our previous work, and when the
central charge $k$ is zero they yield generalizations of the affine HOMFLY-PT
skein category. By exploiting some natural categorical actions of
$\mathcal{H}\textit{eis}_k(A;z,t)$ on generalized cyclotomic quotients, we
prove a basis theorem for morphism spaces.
