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

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