  We compute the group of automorphisms of an arbitrary ind-variety of
(possibly isotropic) generalized flags. Such an ind-variety is a homogeneous
ind-space for one of the ind-groups $SL(\infty)$, $O(\infty)$ or $Sp(\infty)$.
We show that the respective automorphism groups are much larger than
$SL(\infty)$, $O(\infty)$ or $Sp(\infty)$, and present the answer in terms of
Mackey groups. The latter are groups of automorphisms of nondegenerate pairings
of (in general infinite-dimensional) vector spaces. An explicit matrix form of
the automorphism group of an arbitrary ind-variety of generalized flags is also
given. The case of the Sato grassmannian is considered in detail, and its
automorphism group is the projectivization of the connected component of unity
in the group Japanese $GL(\infty)$.
