A note on prime index of a certain subgroup in ${\mathbb{F}}_p^{\ast }$

Under the generalized Riemann hypothesis, we illustrate that the ratio of the set of primes $p$ such that $⟨-1,2⟩$ has an odd prime index in ${\mathbb{F}}_p^{\ast }$ to the set of primes $p$ such that the subgroup has index greater than $2$ nears $46\mathrm{\%}$.