When N ⊂ M is an inclusion of
factors with finite index and a group G acts on N ⊂ M, we compare the standard
invariants of N ⊂ M and the crossed product inclusion N ⋊ G ⊂ M ⋊ G. The cases
when G is a discrete group and when G is a locally compact abelian group are
separately considered. Applying to a common crossed product decomposition, we
obtain comparison results between the type II and type III standard invariants of an
inclusion of type III factors.