  There are four groups $G$ fitting into a short exact sequence $ 1\rightarrow
SL(2,5)\rightarrow G\rightarrow C_2\rightarrow 1, $ where $SL(2,5)$ is the
special linear group of $(2\times 2)$-matrices with entries in the field of
five elements. Except for the direct product of $SL(2,5)$ and $C_2$, there are
two other semidirect products of these two groups and just one non-semidirect
product $SL(2,5).C_2$, considered in this paper. It is known that each finite
nonsolvable group can act on spheres with arbitrary positive number of fixed
points. Clearly, $SL(2,5).C_2$ is a nonsolvable group. Moreover, it turns out
that $SL(2,5).C_2$ possesses a free representation and as such, can potentially
act pseudofreely with nonempty fixed point set on manifolds of arbitrarily
large dimension. We prove that $SL(2,5).C_2$ cannot act effectively with odd
number of fixed points on low-dimensional spheres. In the special case of
effective one fixed point actions, we are able to exclude a wider class of
spheres. Moreover, we prove that specific pseudofree one fixed point actions of
$SL(2,5).C_2$ on spheres do not exist.
