  This project considers the finite symmetry subgroups of the orthogonal group
$\mathrm{O}(3) \subset \mathrm{GL}(3,\mathbb{R})$ and the index $2$
containments $G\lhd \widehat{G}$. The special orthogonal group $\mathrm{SO}(3)
\subset \mathrm{SL}(3,\mathbb{R})$ admits a double cover from the spinor group
$\mathrm{Spin}(3) \cong \mathrm{SU}(2) \subset \mathrm{SL}(2,\mathbb{C})$, and
lifting our subgroups up preserves the network of containments. Those subgroups
not contained in $\mathrm{SO}(3) \subset \mathrm{O}(3) $ are lifted to the
pinor groups $\mathrm{Pin}_{\pm}(3)$ of which there are two choices. For the
index $2$ containments $G\lhd \widehat{G}$, we calculate the Real and complex
Frobenius-Schur indicators, and apply Dyson's classification of antilinear
block structures to produce decorated McKay graphs for each case. We then
explore $KR$-theory as introduced by Atiyah in 1966, which is a variant of
topological $K$-theory for working with topological spaces equipped with an
involution. The GIT quotient spaces $\mathbb{C}^2 // G$, can be equipped by an
involution via the action of $\widehat{G} / G$. In 1983, Gonzalez-Sprinberg and
Verdier showed how one can view the McKay correspondence as an isomorphism
between the $G$-equivariant $K$-theory $K_G(\mathbb{C}^2)$ and the $K$-theory
of the minimal resolution of the singularity $\widetilde{\mathbb{C}^2 // G}$.
We use this to conjecture an analogous a form of the McKay correspondence for
$C_2$-graded groups and $KR$-theory.
