  In earlier works, it was seen that a ${\mathbb Z}/2$ orbifold of the theory
of 24 free two-dimensional chiral fermions admits various sporadic finite
simple groups as global symmetry groups when viewed as an ${\cal N}=1$, ${\cal
N}=2$, or ${\cal N}=4$ superconformal field theory. In this note, we show that
viewing the same theory as an SCFT with extended ${\cal N}=1$ symmetry -- where
the extension is the same one which arises in string compactification on
manifolds of exceptional Spin$(7)$ holonomy -- yields theories which have
global symmetry given by the sporadic groups $M_{24}, Co_2$ or $Co_3$. The
partition functions twined by these symmetries, when decomposed into characters
of the Spin(7) algebra, give rise to two-component vector-valued mock modular
forms encoding an infinite-dimensional module for the corresponding sporadic
groups.
