  Let $X^*$ denote a Banach space with a subsymmetric weak$^*$ Schauder basis
satisfying condition~\eqref{eq:condition-c}. We show that for any operator $T :
X^*\to X^*$, either $T(X^*)$ or $(I-T)(X^*)$ contains a subspace that is
isomorphic to $X^*$ and complemented in $X^*$. Moreover, we prove that
$\ell^p(X^*)$, $1\leq p \leq \infty$ is primary.
