We are motivated by [M. Arkowitz. K. Maruyama. J. Math. Soc. Japan, 66(3):735-743, 2014]: “It would be interesting to compute other Gottlieb groups of Moore spaces such as, for example, Gn+1(M(A, n))” to compute the Gottlieb groups Gn+k(M(Z m +Z2, n)) for k = 1,2 and m ≥1.