Laurent Piriou Université de Nantes, Département de mathématiques 2 rue de la Houssinière BP 92208 Nantes Cedex 03 France laurent.piriou@math.univ-nantes.fr Lionel Schwartz Université Paris 13 Institut Galilée LAGA UMR 7539 du CNRS Av. J. B. Clément 93430 Villetaneuse France schwartz@math.univ-paris13.fr Code AMS 55S10 This article considers two filtrations on the mod-$2$ cohomology $H^*E$ of an abelian $2$-groups $E$. The first one is the primitive fitration, recall that $H^*E$ is a Hopf algebra. The second one is a kind of socle or Loewy filtration of $H^*E$ as unstable module. If dimension of $E$ is $1$ the two filtrations are the same, if the dimension is larger than $2$ it is shown that the filtration are, in some sense compatible. There is an analogous statement in ${\cal F}$, the category of functors from the category of finite dimensional ${\bf F}_2$-vector spaces to the category of all ${\bf F}_2$-vector spaces, for the functor $V \mapsto {\rm map}({\rm Hom}(V,E),{\bf F}_2)$. However, it is better to work with unstable modules because the Steenrod algebra allows computation on certain classes, that are central in the proof, given by the representation theory of symmetric groups that are central in the proof.