Semifield planes admitting the quaternion group $Q_8$

We discuss a well-known conjecture that the full automorphism group of a finite projective plane coordinatized by a semifield is solvable. For a semifield plane of order $p^N$ ($p>2$ is a prime, $4\vert p-1$) admitting an autotopism subgroup $H$ isomorphic to the quaternion group $Q_8$, we construct a matrix representation of $H$ and a regular set of the plane. All nonisomorphic semifield planes of orders $5^4$ and $13^4$ admitting $Q_8$ in the autotopism group are pointed out. It is proved that a semifield plane of order $p^4$, $4\vert p-1$, does not admit $SL(2,5)$ in the autotopism group.