Do $2^{n-1}\equiv1\pmod n$ and $(n-1)/2$ prime imply $n$ prime?

Equivalently: Does $n$ being a Fermat pseudoprimes to base 2 (OEIS A001567) imply that $(n-1)/2$ is composite? That holds for all $n<2^{64}$, based on Jan Feitsma's table.

Motivation is a simplification in the search of safe primes as used in cryptography.

Progress so far: Pocklington's theorem states that if $q>\sqrt n-1$ is a prime dividing $n-1$, and $a^{n-1}\equiv1\pmod n$, then $n$ is prime or $\gcd(a^{(n-1)/q},n)\ne1$. Applying this for $a=2$, it comes that any counterexample $n$ to the propositions would be a multiple of $3$.

The question then boils down to: do $6k+1$ prime imply $4^{6k+1}\not\equiv1\pmod{4k+1}$ ?