Pencils of quadrics in characteristic 2

In order to determine the finite subgroups of the plane Cremona group, one considers automorphisms of certain rational surfaces. A particular example is the del Pezzo surface of degree 4 which is defined by the zeroes of a pair of quadrics in projective 4-space. In characteristic zero the classification is (mostly) complete, but it remains open in positive characteristic.
With this as motivation, we consider normal forms for pencils of quadrics and descriptions of their automorphisms (generalizing the case of a del Pezzo surface of degree 4). We focus on the case of even-dimensional varieties in characteristic 2. (Joint with I. Dolgachev)