  In this article we continue the work started in arXiv:2303.00376v1,
explicitly determining the Weierstrass semigroup at any place and the full
automorphism group of a known $\mathbb{F}_{q^2}$-maximal function field $Y_3$
having the third largest genus, for $q \equiv 1 \pmod 3$. This function field
arises as a Galois subfield of the Hermitian function field, and its uniqueness
(with respect to the value of its genus) is a well-known open problem. Knowing
the Weierstrass semigroups may provide a key towards solving this problem.
Surprisingly enough, $Y_3$ has many different types of Weierstrass semigroups
and the set of its Weierstrass places is much richer than its set of
$\mathbb{F}_{q^2}$-rational places. We show that a similar exceptional
behaviour does not occur in terms of automorphisms, that is,
$\mathrm{Aut}(Y_3)$ is exactly the automorphism group inherited from the
Hermitian function field, apart from small values of $q$.
