  We determine the simple currents and fixed points of the orbifold theory
$CFT\otimes CFT/\mathbb{Z}_2$, given the simple currents and fixed point of the
original $CFT$. We see in detail how this works for the $SU(2)_k$ WZW model,
focusing on the field content (i.e. $h$-spectrum of the primary fields) of the
theory. We also look at the fixed point resolution of the simple current
extended orbifold theory and determine the $S^J$ matrices associated to each
simple current for $SU(2)_2$ and for the $B(n)_1$ and $D(n)_1$ series.
