A prime $p$ is a Sophie Germain prime if its associate $2p + 1$ is also prime.
When is $2$ a primitive root for a Sophie Germain prime $p$ or its associate $2p + 1$?
A previous question found that a Sophie Germain prime $p > 3$ is of the form $6k - 1$: Frank Hubeny (https://math.stackexchange.com/users/312852/frank-hubeny), Show that a Sophie Germain prime $p$ is of the form $6k - 1$ for $p > 3$, URL (version: 2016-02-21): Show that a Sophie Germain prime $p$ is of the form $6k - 1$ for $p > 3$
So far from the discussion there is an answer for the associate prime $2p + 1$. If $p \equiv 1 \pmod{4}$ then $2p + 1 \equiv 3 \pmod{8}$. This implies $2$ is a primitive root modulo $2p + 1$ since that would make $2$ a quadratic nonresidue of $2p + 1$.
Can anything be said about the Sophie Germain prime $p$ itself given that $2p + 1$ is a prime?