Linear autotopism subgroups of semifield projective planes

We investigate the well-known hypothesis of D. R. Hughes that the full collineation group of non-Desarguesian semifield projective plane of a finite order is solvable (the question 11.76 in Kourovka notebook was written down by N. D. Podufalov). This hypothesis is reduced to autotopism group that consists of collineations fixing a triangle. We describe the elements of order 4 and dihedral or quaternion subgroups of order 8 in the linear autotopism group when the semifield plane is of rank 2 over its kernel. The main results can be used as technical for the further studies of the subgroups of even order in an autotopism group for a finite non-Desarguesian semifield plane. The results obtained are useful to investigate the semifield planes with the autotopism subgroups from J. G. Thompson's list of minimal simple groups.