A note on commuting squares arising from automorphisms on subfactors

Using an idea due to Popa, we can associate a commuting square of factors to any given finite set of automorphisms acting on an inclusion of factors of finite index. We use this setting to obtain a simple proof of Popa's classification theorem of strongly outer actions of finitely generated discrete strongly amenable groups on a strongly amenable inclusion of type II1 factors. We also obtain a new complete outer conjugacy invariant for arbitrary automorphisms, which contains the higher obstruction of Kawahigashi and the standard invariant as a special case.