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Based on the fact that, for a subfactor $N$ of a II$_1$ factor $M,$ the firstnon-trivial Jones index is 2 and then $M$ is decomposed as the crossed productof $N$ by an outer action of ${\mathbb{Z}}_2,$ we study pairs $ \{N, uNu^* \}$from a view point of entropy for two subalgebras of $M$ with a connection tothe entropy for automorphisms, where the inclusion of II$_1$ factors $ N\subset M$ is given as $M$ is the crossed product of $N$ by a finite group ofouter automorphisms and $u$ is a unitary in $M.$

Conjugate Pairs of Subfactors and Entropy for Automorphisms