  Let $\mathbb{F}_q$ denote a finite field of odd cardinality $q$,
$\mathbb{A}=\mathbb{F}_q[T]$ the polynomial ring over $\mathbb{F}_q$ and
$k=\mathbb{F}_q(T)$ the rational function field over $\mathbb{F}_q$. In this
paper, we compute the average value of the size of the group
$K_2(\mathcal{O}_{\gamma D})$, where $\mathcal{O}_{\gamma D}$ denotes the
integral closure of $\mathbb{A}$ in $k(\sqrt{\gamma D})$, $D$ is a monic,
square-free polynomial of even degree and $\gamma$ is a fixed generator of
$\mathbb{F}_q^*$.
