We introduce a notion of a pentagonal equation for an 
adjointable operator on a Hilbert $C^*$-module 
in full generality.  
We call a unitary operator on a Hilbert $C^*$-module 
a multiplicative unitary operator (MUO) when it 
satisfies the pentagonal equation.  
We give a sufficient condition for the existence of 
an MUO associated with a general inclusion of $C^*$-algebras.  
Then  we study an MUO when the inclusion is of index-finite 
type in the sense of Watatani.  We also give an explicit 
formula for the MUO when the inclusion arises from 
a crossed product of a $C^*$-algebra by a finite group.