the theory reduces to Einstein gravity in vacuum, imposing that the solution is Schwarzschild, or more generically, Kerr-Newman family. We therefore show the non-existence of hairy black holes. But in 1972, a conformally coupled scalar field to GR giving rise to solution asymptotically flat and different form Schwarzschild was derived. The scalar field is still singular on the horizon but without a divergence of the curvature tensor. The action is defined as $$\begin{aligned}
S=\int{\rm d}^4x\sqrt{-g}\Bigl[\frac{R}{2}-\frac{1}{2}(\partial\phi)^2-\frac{R}{12}\phi^2\Bigr]\,,\end{aligned}$$ where the solution is $$\begin{aligned}
ds^2 &=-\Bigl(1-\frac{M}{r}\Bigr)^2 dt^2+\frac{dr^2}{\Bigl(1-\frac{M}{r}\Bigr)^2}+r^2d\Omega^2\,,\\
\phi &=\frac{\sqrt{6}M}{r-M}\,.\end{aligned}$$ This is known as the Bocharova-Bronnikov-Melnikov-Bekenstein (BBMB) solution [@Bocharova:1970skc; @Bekenstein:1974sf]. Even if the solution is not Schwarzschild, it is the extreme Reissner-Nordström solution, there are no other parameters to describe the solution except the mass. Therefore it is not a hairy black hole solution (or primary hair) but secondary hair solution [@Herdeiro:2015waa].

The stability of this solution is unclear since both situations have been claimed$-$stable in [@McFadden:2004ni] and unstable in [@Bronnikov:1978mx]. Unfortunately, the stability has been studied only for radial (monopole) perturbations in case of even-parity perturbations. We, therefore, extend the analysis and study completely the odd-parity perturbations. In our notations, for this case, we have $$K=X,~G_3=0,~G_4=\frac{1}{2}\Bigl(1-\frac{\phi^2}{6}\Bigr),~G_5=0\,.$$ Therefore, we obtain $${\cal F}={\cal G}={\cal H}=1-\frac{\phi^2}{6}=\frac{r(r-2M)}{(r-M)^2}\,.$$ It 