  Let k=F_q be a finite field of characteristic 2. A genus 3 curve C/k has many
involutions if the group of k-automorphisms admits a C_2\times C_2 subgroup H
(not containing the hyperelliptic involution if C is hyperelliptic). Then C is
an Artin-Schreier cover of the three elliptic curves obtained as the quotient
of C by the nontrivial involutions of H, and the Jacobian of C is k-isogenous
to the product of these three elliptic curves. In this paper we exhibit
explicit models for genus 3 curves with many involutions, and we compute
explicit equations for the elliptic quotients. We then characterize when a
triple (E_1,E_2,E_3) of elliptic curves admits an Artin-Schreier cover by a
genus 3 curve, and we apply this result to the construction of maximal curves.
As a consequence, when q is nonsquare and m=\lfloor 2 sqrt(q) \rfloor = 1,5,7
mod 8, we obtain that N_q(3)=1+q+3m. We also show that this occurs for an
infinite number of values of q nonsquare.
