The $$\mathbb {Z}_{2^s}$$ Z 2 s -additive codes are subgroups of $$\mathbb {Z}^n_{2^s}$$ Z 2 s n , and can be seen as a generalization of linear codes over $$\mathbb {Z}_2$$ Z 2 and $$\mathbb {Z}_4$$ Z 4 . A $$\mathbb {Z}_{2^s}$$ Z 2 s -linear Hadamard code is a binary Hadamard code which is the Gray map image of a $$\mathbb {Z}_{2^s}$$ Z 2 s -additive code. It is known that the dimension of the kernel can be used to give a complete classification of the $$\mathbb {Z}_4$$ Z 4 -linear Hadamard codes. In this paper, the kernel of $$\mathbb {Z}_{2^s}$$ Z 2 s -linear Hadamard codes of length $$2^t$$ 2 t and its dimension are established for $$s > 2$$ s > 2 . Moreover, we prove that this invariant only provides a complete classification for some values of t and s. The exact amount of nonequivalent such codes are given up to $$t=11$$ t = 11 for any $$s\ge 2$$ s ≥ 2 , by using also the rank.