  This paper is concerned with the Smith question which reads as follows. Is it
true that for a finite group acting smoothly on a sphere with exactly two fixed
points, the tangent spaces at the fixed points have always isomorphic group
module structures defined by differentiation of the action? We show that one
can answer this question negatively by using the technique of induction of
group representations. We apply our results to indicate new dimensions of
spheres admitting actions of specific Oliver groups, which give the negative
answer to the Smith question. In particular, for the first time, we indicate
some solvable non-nilpotent Oliver groups which yield negative answers to the
Smith question.
