AbstractIn previous work, the authors defined a category $$\text {SMod}_F$$ SMod F of finite Galois modules decorated with local conditions for each global field F. In this paper, given an extension K/F of global fields, we define a restriction of scalars functor from $$\text {SMod}_K$$ SMod K to $$\text {SMod}_F$$ SMod F and show that it behaves well with respect to the Cassels–Tate pairing. We apply this work to study the class groups of global fields in the context of the Cohen–Lenstra heuristics.