A semifield plane of rank 2 admitting nonlinear Baire involution

We obtain a matrix representation of the regular set of a semifield plane of order $q^4$ with a kernel of order $q^2$ ($q=2^k$), where the semifield plane admits semilinear Baire involution. The possibility of coexistence of linear and semilinear Baire involutions in the group of autotopisms of a plane is studied.