Transfer kernels and the Schur multiplier

In this article we prove that if $H$ is a normal subgroup of a finite group $G$ with $G/H$ cyclic, then the kernel of the transfer map $G/G' \longrightarrow H/H'$ equals $(G:H)$ times the order of the cokernel of the natural map $M(H) \longrightarrow M(G)$.