On different families of invariant irreducible polynomials over F2

Using a natural action of the permutation group S3 on the set of irreducible polynomials, we attach to each subgroup of S3 the family of its invariant polynomials. Enumeration formulas for the trivial subgroup and for one transposition subgroup were given by Gauss (1863) (for prime fields) [1], and Carlitz (1967) (for all finite base fields) [2], . Respectively, they allow to enumerate all irreducible and self-reciprocal irreducible polynomials. In our context, the last remaining case concerned the alternating subgroup A3. We give here the corresponding enumeration formula restricted to F2 base field. We wish this will give an interesting basis for subsequent developments analogous to those of Meyn (1990) [3], and Cohen (1992) [4].