In the question I ask about one elliptic fibration on the surface $$ K\!: y^2 + x_1x_2y = (x_1x_2)^2(x_1 + x_2 + 1) + (x_1 + x_2)^2. $$ over a finite field $\mathbb{F}_q$ of characteristic $2$ such that $2 \nmid \log_2 q$. This is a model of the Kummer surface of $E \!\times\! E^\prime$, i.e., the quotient $E \!\times\! E^\prime/[-1]$, where the elliptic curves $$ E\!:y_1^2 + x_1y_1 = x_1^3 + 1,\qquad E^\prime\!: y_2^2 + x_2y_2 = x_2^3 + x_2^2 + 1 $$ are the quadratic twists to each other. Do you see other elliptic $\mathbb{F}_q$-fibrations on $K$ ?
Thanks in advance.
