Let $p$ be a prime number and $S_p=\{(n!)^2 \bmod p, n=1,2,\dotsc,p-1\}$ the set of residues mod $p$ of squares of factorials. This set is obviously a subset of the group of quadratic residues mod p. For $p=3,5,7,13,17,23,29$ it is also a group for multiplication mod $p$, i.e. a subgroup of the group of quadratic residues.
Question: Are there infinitely many primes $p$ for which $S_p$ is a group modulo $p$?