  We establish necessary and sufficient conditions for the stability of the
finite section method for operators belonging to a certain $C^*$-algebra of
operators acting on the Hilbert space $l^2_H(\mathbb{Z})$ of $H$-valued
sequences where $H$ is a given Hilbert space. Identifying $l^2_H(\mathbb{Z})$
with the $L^2_H$-space over the unit circle, the $C^*$-algebra in question is
the one which contains all singular integral operators with flip and piecewise
quasicontinous $\mathcal{L}(H)$-valued generating functions on the unit circle.
The result is a generalization of an older result where the same problem, but
without the flip operator was considered. The stability criterion is obtained
via $C^*$-algebra methods and says that a sequence of finite sections is stable
if and only if certain operators associated with that sequence (via
$^*$-homomorphisms) are invertible.
