On Gottlieb groups Gₙ₊ₖ(M(Z^m + Z_2,n)) for k=1,2

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, G{n+1}(M(A,n))" to compute the Gottlieb groups Gn+k(M(ℤm⊕ℤ2,n)) for k=1,2 and m≥1.