Summary: We propose a test for testing the equality of the mean vectors of two groups with unequal covariance matrices based on \(N_{1}\) and \(N_{2}\) independently distributed \(p\)-dimensional observation vectors. It will be assumed that \(N_{1}\) observation vectors from the first group are normally distributed with mean vector \(\mathbf {\mu}_1\) and covariance matrix \(\mathbf {\Sigma}_1\). Similarly, the \(N_{2}\) observation vectors from the second group are normally distributed with mean vector \(\mathbf {\mu}_2\) and covariance matrix \(\mathbf {\Sigma}_2\). We propose a test for testing the hypothesis that \(\mathbf {\mu}_1=\mathbf {\mu}_2\). This test is invariant under the group of \(p\times p\) nonsingular diagonal matrices. The asymptotic distribution is obtained as \((N_{1},N_{2},p)\to \infty \) and \(N_{1}/(N_{1}+N_{2})\to k\in (0,1)\) but \(N_{1}/p\) and \(N_{2}/p\) may go to zero or infinity. It is compared with a recently proposed non-invariant test. It is shown that the proposed test performs the best.