Summary: We prove in this paper some sharp weighted inequalities for the vector-valued maximal function \(\overline M_q\) of Fefferman and Stein defined by \[ \overline M_qf(x)=\left(\sum_{i=1}^{\infty}(Mf_i(x))^{q}\right)^{1/q}, \] where \(M\) is the Hardy-Littlewood maximal function. As a consequence we derive the main result establishing that in the range \(1<q<p<\infty\) there exists a constant \(C\) such that \[ \int_{\mathbf{R}^{n}}\overline M_qf(x)^p w(x) dx\leq C \int_{\mathbf{R}^n}|f(x)|^{p}_{q} M^{[\frac pq]+1}w(x) dx. \] Furthermore, the result is sharp since \(M^{[\frac pq]+1}\) cannot be replaced by \(M^{[\frac pq]}\). We also show the following endpoint estimate \[ w(\{x\in \mathbf{R}^n:\overline M_qf(x)>\lambda\}) \leq \frac C\lambda \int_{\mathbf{R}^n} |f(x)|_q Mw(x) dx, \] where \(C\) is a constant independent of \(\lambda\).