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We discuss an isomorphism between the possible anomalies of$(d+1)$-dimensional quantum field theories with $\mathbb{Z}_{2}$ unitary globalsymmetry, and those of $d$-dimensional quantum field theories withtime-reversal symmetry $\mathsf{T}$. This correspondence is an instance ofsymmetry defect decoration. The worldvolume of a $\mathbb{Z}_{2}$ symmetrydefect is naturally invariant under $\mathsf{T},$ and bulk $\mathbb{Z}_{2}$anomalies descend to $\mathsf{T}$ anomalies on these defects. We illustratethis correspondence in detail for $(1+1)d$ bosonic systems where the bulk$\mathbb{Z}_{2}$ anomaly leads to a Kramers degeneracy in the symmetry defectHilbert space, and exhibit examples. We also discuss $(1+1)d$ fermion systemsprotected by $\mathbb{Z}_{2}$ global symmetry where interactions lead to a$\mathbb{Z}_{8}$ classification of anomalies. Under the correspondence, this isdirectly related to the $\mathbb{Z}_{8}$ classification of $(0+1)d$ fermionsprotected by $\mathsf{T}$. Finally, we consider $(3+1)d$ bosonic systems with$\mathbb{Z}_{2}$ symmetry where the possible anomalies are classified by$\mathbb{Z}_{2}\times \mathbb{Z}_{2}$. We construct topological field theoriesrealizing these anomalies and show that their associated symmetry defectssupport anyons that can be either fermions or Kramers doublets.

Decorated $\mathbb{Z}_{2}$ Symmetry Defects and Their Time-Reversal Anomalies