It seems to me that the characteristic of the field does not play a big role in this question: here is a sketch of an argument.

Note first of all, that all the invariants are linear combinations of "symmetrized monomials": if *m* is a monomial in the polynomial ring, then form the sum of all the translates of *m* by the elements of your group.  This means that every invariant in the polynomial ring comes from an invariant polynomial with coefficients in the prime field $\mathbb{F}_2$ of *K* and that invariants with coefficients in $\mathbb{F}_2$ are the reduction of invariants with integer coefficients.  Thus we have translated the question over characteristic two to a question over the integers: it suffices to find generators and relations for the ring of invariants of your group over the integers to find generators and relations in any ring.

Over the integers I do not know what the answer is, but if you do know what the answer is over any field of characteristic different from two, maybe you can now fill in the argument.  Thinking briefly about the set of generators, it seems like you might simply need the "symmetrized square-free monomials", with relations that are a bit tedious to write down, but that maybe can be nicely interpreted.