The work of R. Gannouji is partially supported by DII-PUCV No 039.450/2017 and Fondecyt project No 1171384. M. Gonzalez-Espinoza acknowledges support from a PUCV doctoral scholarship.

Verification of the paper [@Kobayashi:2012kh] {#appenA}
=============================================

In [@Kobayashi:2012kh], the authors performed an analysis of black hole perturbations in the Horndeski model under odd-parity perturbations. In this appendix, we will show that in the simplest case of Schwarzschild solution in GR, their result do not produce the famous Regge and Wheeler potential.

Considering GR and therefore, ${\cal F}={\cal G}={\cal H}=1$ and $A=B=1-2M/r$, the second order action they found gives $$\begin{aligned}
S\propto \int{\rm d}t{\rm d}r\Bigl[\frac{\dot Q^2}{A^2}-Q'^2-\frac{l(l+1)}{r^2A}Q^2-VQ^2\Bigr]\,,\end{aligned}$$ where $$\begin{aligned}
V(r)=\frac{M(61M-32r)}{4r^2(r-2M)^2}\,.\end{aligned}$$ Transforming the coordinate $r$ to the tortoise coordinate $dr=A dr^*$ and considering the change of variable $Q=\sqrt{A}\xi$, we get after an integration by parts $$\begin{aligned}
S\propto \int{\rm d}t{\rm d}r^*\Bigl[\dot\xi^2-\Bigl(\frac{d\xi}{dr^*}\Bigr)^2-W\xi^2\Bigr]\,,\end{aligned}$$ which gives after variation $$\begin{aligned}
-\frac{\partial^2 \xi}{\partial t^2}+\frac{\partial^2 \xi}{\partial r^{*2}}-W\xi=0\,,\end{aligned}$$ where $$\begin{aligned}
W(r)=A\Bigl[\frac{l(l+1)}{r^2}+\frac{M(41M-24r)}{4r^3(r-2M)}\Bigr]\,,\end{aligned}$$ which is clearly not the Regge-Wheeler potential.

Finally, it can easily be checked that our result is generalizing previous calculations derived for specific models (see e.g. [@Anabalon:2014lea; @Cisterna:2015uya]).

Background equations {#appenB}
====================

Here we define the quantities that appear in the background equations ${\cal E}_A={\cal E}_B={\cal E}_C=0$

$$\begin{aligned}
{\cal E}_A&:=&K+B\phi'X'G_{3X}-2XG_{3\phi}
+
\frac{2}{C}\left(1+\dfrac{BC'^2}{4C}-BC''-\dfrac{B'C'}{2}\right)  G_4
+
