  The behavior of $b=2$ real-space renormalization group (RSRG) maps like the
majority rule and the decimation map was examined by numerically applying RSRG
steps to critical $q=2,3,4$ Potts spin configurations. While the majority rule
is generally believed to work well, a more thorough investigation of the action
of the map has yet to be considered in the literature. When fixing the size of
the renormalized lattice $L_g$ and allowing the source configuration size $L_0$
to vary, we observed that the RG flow of the spin and energy correlation under
the majority rule map appear to converge to a nontrivial model-dependent curve.
We denote this property as "faithfulness", because it implies that some
information remains preserved by RSRG maps that fall under this class.
Furthermore, we show that $b=2$ weighted majority-like RSRG maps acting on the
$q=2$ Potts model can be divided into two categories, maps that behave like
decimation and maps that behave like the majority rule.
