  A Banach space $X$ is said to be Q.H.I. if eve\-ry infinite dimensional
quo\-tient spa\-ce of $X$ is H.I.: that is, a space is Q.H.I. if the H.I.
property is not only stable passing to subspaces, but also passing to quotients
and to the dual. We show that Gowers-Maurey's space is Q.H.I.; then we provide
an example of a reflexive H.I. space ${\cal X}$ whose dual is not H.I., from
which it follows that $\cal X$ is not Q.H.I.
