A $Z_2Z_4$-linear Hadamard code of length $α+2β=2^t$ is a binary Hadamard code which is the Gray map image of a $Z_2Z_4$-additive code with $α$ binary coordinates and $β$ quaternary coordinates. It is known that there are exactly $[(t-1)/2]$ and $[t/2]$ nonequivalent $Z_2Z_4$-linear Hadamard codes of length $2^t$, with $α=0$ and $α\not=0$, respectively, for all $t\geq 3$. In this paper, it is shown that each $Z_2Z_4$-linear Hadamard code with $α=0$ is equivalent to a $Z_2Z_4$-linear Hadamard code with $α\not=0$; so there are only $[t/2]$ nonequivalent $Z_2Z_4$-linear Hadamard codes of length $2^t$. Moreover, the order of the monomial automorphism group for the $Z_2Z_4$-additive Hadamard codes and the permutation automorphism group of the corresponding $Z_2Z_4$-linear Hadamard codes are given.