  In this paper we examine bases for finite index inclusion of $II_1$ factors
and connected inclusion of finite dimensional $C^*$- algebras. These bases
behave nicely with respect to basic construction towers. As applications we
have studied automorphisms of the hyperfinite $II_1$ factor $R$ which are
`compatible with respect to the Jones' tower of finite dimensional
$C^*$-algebras'. As a further application, in both cases we obtain a
characterization, in terms of bases, of basic constructions. Finally we use
these bases to describe the phenomenon of multistep basic constructions (in
both the cases).
