  If a finite group action $\alpha$ on a unital $C^*$-algebra $M$ is saturated,
the canonical conditional expectation $E:M\to M^\alpha$ onto the fixed point
algebra is known to be of index finite type with $Index(E)=|G|$ in the sense of
Watatani. More generally if a finite dimensional Hopf $*$-algebra $A$ acts on
$M$ and the action is saturated, the same is true with $Index (E)=\dim(A)$. In
this paper we prove that the converse is true. Especially in case $M$ is a
commutative $C^*$-algebra $C(X)$ and $\alpha$ is a finite group action, we give
an equivalent condition in order that the expectation $E:C(X)\to C(X)^\alpha$
is of index finite type, from which we obtain that $\alpha$ is saturated if and
only if $G$ acts freely on $X$.
  Actions by compact groups are also considered to show that the gauge action
$\gamma$ on a graph $C^*$-algebra $C^*(E)$ associated with a locally finite
directed graph $E$ is saturated.
