A maj-inv bijection for C₂ wr A_n

We give a bijective proof of the MacMahon-type equidistribution over the group of signed even permutations C_2 \wr A_n that was stated in [Bernstein. Electron. J. Combin. 11 (2004) 83]. This is done by generalizing the bijection that was introduced in the bijective proof of the equidistribution over the alternating group A_n in [Bernstein and Regev. Sem. Lothar. Combin. 53 (2005) B53b].