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Let [Special characters omitted.] ( n ) be the subalgebra of the mod-2 Steenrod algebra generated by {1, Sq ¹ ,..., Sq ^{2 n} }. I hope to construct a 2-group G with the property H *( G , [Special characters omitted.] /2) [congruent with] [Special characters omitted.] ([Special characters omitted.] /2, [Special characters omitted.] /2). The difficulty in this construction is due to the different actions of Sq ⁰ on the cohomology of a group and the cohomology of a Hopf algebra.

I use the May spectral sequence to calculate [Special characters omitted.] ([Special characters omitted.] /2, [Special characters omitted.] /2). I then use central extensions of groups corresponding to some of the May spectral sequence differentials. I use the Lyndon-Hochschild-Serre (LHS) spectral sequence to calculate the group cohomology for each central extension, and also maps to the LHS spectral sequences for certain subgroups. I use additional central group extensions for each time the Sq ⁰ action on the May spectral sequence differs from the Sq ⁰ action on the LHS spectral sequence.

This process works for realizing some of the May spectral sequence differentials in the cohomology of groups, but not all. In particular, there are two differentials in the calculation of [Special characters omitted.] ([Special characters omitted.] /2, [Special characters omitted.] /2) that cannot be constructed in this way. Thus, the result of Sq ⁰ acting on [Special characters omitted.] ([Special characters omitted.] /2, [Special characters omitted.] /2) cannot necessarily be accomplished using extra central extensions of groups.