Is there any existing standard name for primes of the following kind in mathematics?

Question

Given a prime $p$ all primes $q$ such that $$q \bmod p = 1\text{ and }2^{(q-1)/p} \bmod q = 1$$Do primes $q$ have any standard name ?

Answer 1

I don't think that there is a special name for primes like these. Note however that the condition $$2^{(q-1)/p} \equiv 1 \pmod q$$ means that the order of $2$ in $(\mathbb Z/q\mathbb Z)^* = \mathbb Z/(q-1)\mathbb Z$ is a proper divisor of $q-1$. In other words, your condition means that $2$ does not generate $(\mathbb Z/q\mathbb Z)^*$.

This, for instance, excludes all Fermat primes: if $q = 2^{2^n} + 1$, then necessarily $p = 2$, so that $1 < 2^{(q-1)/p} = 2^{2^{n-1}} < 2^{2^n}$. In particular

$$2^{(q-1)/p} = 2^{2^{n-1}} \not\equiv 1 \pmod{2^{2^n}+1}.$$

So Fermat primes never satisfy what you want. However, this does not mean that if $q$ is not Fermat, then $q$ satisfies your condition ($q = 19$ is a counterexample).

You can have a look at the the table on Wikipedia, which lists generators of $(\mathbb Z/n \mathbb Z)^*$. Just have a look at the rows where $n$ is prime and $2$ is not a generator.