symmetric monoidal (∞,1)-category of spectra
A $\beta$-ring is a commutative ring, $R$, equipped with a set of operations, $\beta_H: R \to R$, indexed by subgroups of symmetric groups, $S_n$, satisfying a number of conditions. They may be seen as collections of integral linear combinations of generalized symmetric powers defined on Burnside rings. The cohomotopy of a space, $\pi^0(X)$, is a $\beta$-ring (Guillot 06, Thrm 4.5).
They are completely unrelated to relational beta-modules.
Note that there are variations in the literature as to the definition of $\beta$-rings. For a close comparison with λ-rings, see
Other references
Ernesto Vallejo, The free $\beta$-ring on one generator, Journal of Pure and Applied Algebra 86(1), 1993, pp. 95-108, (doi)
N.W. Rymer, Power operations on the Burnside ring, J. London Math. Sot. (2) 15 (1977) 75-80.
E. Vallejo, Polynomial operations from Burnside rings to representation functors, J. Pure Appl. Algebra 65 (1990) 163-190.
G. Ochoa, Outer plethysm, Burnside rings and $\beta$-rings, J. Pure Appl. Algebra 55 (1988), 173-195.
I. Morris and C.D. Wensley, Adams operations and λ-operations in β-rings, Discrete Mathematics Volume 50, 1984, Pages 253-270, (doi)