  The $\mathbb{Z}_{2^s}$-additive codes are subgroups of $\mathbb{Z}^n_{2^s}$,
and can be seen as a generalization of linear codes over $\mathbb{Z}_2$ and
$\mathbb{Z}_4$. A $\mathbb{Z}_{2^s}$-linear Hadamard code is a binary Hadamard
code which is the Gray map image of a $\mathbb{Z}_{2^s}$-additive code. It is
known that the dimension of the kernel can be used to give a complete
classification of the $\mathbb{Z}_4$-linear Hadamard codes. In this paper, the
kernel of $\mathbb{Z}_{2^s}$-linear Hadamard codes and its dimension are
established for $s > 2$. 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 $t=11$ for any $s\geq 2$, by using
also the rank and, in some cases, further computations.
