Projective symmetries of Chebyshev polynomials

Let $k$ be an algebraically closed field of odd characteristic $p$, and let $φ_d$ be the monic Chebyshev polynomial defined by $φ_d(t+t^{-1})=t^d+t^{-d}$. For $d>4$ with $p\nmid d$, let $H_d\le\text{PGL}_2(k)$ be the stabilizer of the zero set of $φ_d$. We show that $H_d\cong \begin{cases} D_d,&\text{if }2d=p^r+1\;\text{ for some integer }\;r\geq 1,\\ S_3,&\text{if }4d=p^r+1\;\text{ for some integer }\; r\geq 1,\\ C_2,&\text{otherwise}. \end{cases}$