  We make a detailed study of locally inner actions on C*-algebras whose
primitive ideal spaces have locally compact Hausdorff complete regularizations.
We suppose that $G$ has a representation group and compactly generated
abelianization $G_{ab}$. Then if the complete regularization of $\Prim(A)$ is
$X$, we show that the collection of exterior equivalence classes of locally
inner actions of $G$ on $A$ is parameterized by the group $\E_G(X)$ of exterior
equivalence classes of $C_0(X)-actions of $G$ on $C_0(X,\K)$. Furthermore, we
exhibit a group isomorphism of $\E_G(X)$ with the direct sum $H^1(X,\sheaf
\hat{G_{ab}}) \oplus C(X,H^2(G,\T))$. As a consequence, we can compute the
equivariant Brauer group $\Br_G(X)$ for $G$ acting trivially on $X$.
