  Given a discrete quantum group $H$ with a finite normal quantum subgroup $G$,
we show that any positive, possibly unbounded, harmonic function on $H$ with
respect to an irreducible invariant random walk is $G$-invariant. This implies
that, under suitable assumptions, the Poisson and Martin boundaries of $H$
coincide with those of $H/G$. A similar result is also proved in the setting of
exact sequences of C$^*$-tensor categories. As an immediate application, we
conclude that the boundaries of the duals of the group-theoretical easy quantum
groups are classical.
