Double affine Frobenius Hecke algebras

We define double affine Hecke algebras associated to the ${\text{GL}}_n$ and ${\text{SL}}_n$ root data, decorated by a supersymmetric Frobenius superalgebra. When the Frobenius algebra is the group algebra of a finite group, we obtain new double affine versions of the quantum wreath Hecke algebras, and in particular of the affine Yokonuma-Hecke algebras. We prove a basis theorem, describe the centre for generic parameters, and define polynomial representations. We also provide a diagrammatic interpretation, from which we deduce certain involutions and symmetries of the algebra, and use it to describe an ${\text{SL}}_2(\mathbb{Z})$-action by outer automorphisms.