Are there any known Fermat pseudoprimes $p\;$ to base $2\;$ (Sarrus or Poulet numbers) with the properties $q = (p-1)/2\;$ is prime and $p \equiv 0 \pmod 3?$
I was not able to find any example up to $p = 2^{31}-1.$ Is there an argument that they cannot exist?
This question is related to Fast check of safe primes or Sophie Germain primes where I added the condition $p \not \equiv 0 \pmod 3?$