This paper contains results on the structure of the group,
Diff
G
r
(
M
)
\operatorname {Diff} _G^r(M)
, of equivariant
C
r
{C^r}
-diffeomorphisms of a free action of the compact Lie group
G
G
on
M
M
.
Diff
G
r
(
M
)
\operatorname {Diff} _G^r(M)
is shown to be a locally trivial principal bundle over a submanifold of
Diff
r
(
X
)
,
X
{\operatorname {Diff} ^r}(X),X
the orbit manifold. The structural group of this bundle is
E
r
(
G
,
M
)
{E^r}(G,M)
, the set of equivariant
C
r
{C^r}
-diffeomorphisms which induce the identity on
X
X
.
E
r
(
G
,
M
)
{E^r}(G,M)
is shown to be a submanifold of
Diff
r
(
M
)
{\operatorname {Diff} ^r}(M)
and in fact a Banach Lie group
(
r
>
∞
)
(r > \infty )
.