Let $S$ be a subset of $\{0,1\}^n$ such that any two elements of $S$ are at least (Hamming) distance 5 apart. I'm looking for an upper bound on the size of the automorphism group of $S$.

There's a trivial upper bound of $2^nn!$ (the number of automorphisms of $\{0,1\}^n$), and an easy lower bound of $2^{n/5}(n/5)!$ (take S to be all elements of the form $xxxxx$, where $x$ is a bitstring of length $n/5$).

Any bound of the form $n!/n^{cn}$ for any $c>0$ would be helpful.