In 1960, P.A. Smith raised an isomorphism problem.
Is it true that the tangential $G$-modules at two fixed points
of an arbitrary smooth $G$-action on a sphere with exactly
two fixed points are isomorphic to each other? Given a finite
group, the Smith set of the group means the subset of real
representation ring consisting of all differences of Smith
equivalent representations. Many researchers have studied
the Smith equivalence for various finite groups. But the
Smith sets for non-perfect groups were rarely determined.
In particular, the Smith set for a non-gap group has not been
determined unless it is trivial. We determine the Smith set
for the non-gap group $G = S_{5} \times C_{2} \times \dots
\times C_{2}$.