We assume that a factor is equipped with outer actions of two finite groups, and consider a factor-subfactor pair consisting of the crossed product by one group and the fixed point algebra by the other. For this inclusion of factors, the group of one-dimensional bimodules appearing at even levels of the associated graph and that of non-strongly outer automorphisms for the subfactor (i.e., centrally trivial ones in the strongly amenable case) are determined.
Jeong Hee HONG. Hideki KOSAKI. "The group of one-dimensional bimodules arising from composition of subfactors." J. Math. Soc. Japan 52 (2) 293 - 333, April, 2000. https://doi.org/10.2969/jmsj/05220293