  Let $A$ be the ring of elements in an algebraic function field $K$ over
$\mathbb{F}_q$ which are integral outside a fixed place $\infty$. In contrast
to the classical modular group $SL_2(\mathbb{Z})$ and the Bianchi groups, the
{\it Drinfeld modular group} $G=GL_2(A)$ is not finitely generated and its
automorphism group $\mathrm{Aut}(G)$ is uncountable. Except for the simplest
case $A=\mathbb{F}_q[t]$ not much is known about the generators of
$\mathrm{Aut}(G)$ or even its structure. We find a set of generators of
$\mathrm{Aut}(G)$ for a new case. \par On the way, we show that {\it every}
automorphism of $G$ acts on both, the {\it cusps} and the {\it elliptic points}
of $G$. Generalizing a result of Reiner for $A=\mathbb{F}_q[t]$ we describe for
each cusp an uncountable subgroup of $\mathrm{Aut}(G)$ whose action on $G$ is
essentially defined on the stabilizer of that cusp. In the case where $\delta$
(the degree of $\infty$) is $1$, the elliptic points are related to the
isolated vertices of the quotient graph $G\setminus\mathcal{T}$ of the
Bruhat-Tits tree. We construct an infinite group of automorphisms of $G$ which
fully permutes the isolated vertices with cyclic stabilizer.
