Wilson's theorem modulo higher prime powers III: The cases modulo $p^6$ and $p^7$

Extending previous work of the author, we compute the Wilson quotient modulo $p^5$ and $p^6$, and equivalently $(p-1)!$ modulo $p^6$ and $p^7$, respectively. Further, we determine some power sums of the Fermat quotients up to modulo $p^6$. Subsequently, we discuss some patterns that occur in the $p$-adic coefficients of the Wilson quotient as well as of $(p-1)!$, whereby the original congruence $(p-1)!\equiv -1(\mathrm{m}\mathrm{o}\mathrm{d}p)$ fits perfectly into the theory.