  We study linear perturbations about static and spherically symmetric black
hole solutions with stealth scalar hair in degenerate higher-order
scalar-tensor (DHOST) theories. We clarify master variables and derive the
quadratic Lagrangian for both odd- and even-parity perturbations. It is shown
that the even modes are in general plagued by gradient instabilities, or
otherwise the perturbations would be strongly coupled. Several possible ways
out are also discussed.
