  Let $S$ be an unramified regular local ring of mixed characteristic two and
$R$ the integral closure of $S$ in a biquadratic extension of its quotient
field obtained by adjoining roots of sufficiently general square free elements
$f,g\in S$. Let $S^2$ denote the subring of $S$ obtained by lifting to $S$ the
image of the Frobenius map on $S/2S$. When at least one of $f,g\in S^2$, we
characterize the Cohen-Macaulayness of $R$ and show that $R$ admits a
birational small Cohen-Macaulay module. It is noted that $R$ is not
automatically Cohen-Macaulay in case $f,g\in S^2$ or if $f,g\notin S^2$.
