  Let $\mathbb{F}_q$ be the finite field of order $q$ and $F=\mathbb{F}_q(x)$
the rational function field. In this paper, we give a characterization of the
cyclotomic function fields $F(\Lambda_M)$ with modulus $M$, where $M \in
\mathbb{F}_q[T]$ is a monic and irreducible polynomial of degree two. We also
provide the full automorphism group of $F(\Lambda_M)$ in odd characteristic,
extending results of \cite{MXY2016} where the automorphism group of
$F(\Lambda_M)$ over $\mathbb{F}_q$ was computed.
