  We discuss an isomorphism between the possible anomalies of
$(d+1)$-dimensional quantum field theories with $\mathbb{Z}_{2}$ unitary global
symmetry, and those of $d$-dimensional quantum field theories with
time-reversal symmetry $\mathsf{T}$. This correspondence is an instance of
symmetry defect decoration. The worldvolume of a $\mathbb{Z}_{2}$ symmetry
defect is naturally invariant under $\mathsf{T},$ and bulk $\mathbb{Z}_{2}$
anomalies descend to $\mathsf{T}$ anomalies on these defects. We illustrate
this correspondence in detail for $(1+1)d$ bosonic systems where the bulk
$\mathbb{Z}_{2}$ anomaly leads to a Kramers degeneracy in the symmetry defect
Hilbert space, and exhibit examples. We also discuss $(1+1)d$ fermion systems
protected by $\mathbb{Z}_{2}$ global symmetry where interactions lead to a
$\mathbb{Z}_{8}$ classification of anomalies. Under the correspondence, this is
directly related to the $\mathbb{Z}_{8}$ classification of $(0+1)d$ fermions
protected 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 theories
realizing these anomalies and show that their associated symmetry defects
support anyons that can be either fermions or Kramers doublets.
