Let $G$ be a finite group, and let $M$ be a finitely generated $G$-module, that is, a finitely generated abelian group on which $G$ acts. We work with the first homology group $$ H_1(G,M).$$

For any cyclic subgroup $C\subseteq G$, we consider the inclusion map $i_C\colon C\hookrightarrow G$ and the induced homomorphism $$ i_{C,*} \colon\, H_1(C,M)\to H_1(G,M).$$

Following Sansuc's paper of 1981, we say that $G$ is metacyclic if all its Sylow subgroups are cyclic. For example, the symmetric group $S_3$ is metacyclic.

Question. Is it true that if $G$ is metacyclic in the sense of Sansuc, then the images $$ {\rm im}\big[i_{C,*} \colon H_1(C,M)\to H_1(G,M)\big]$$ for all cyclic subgroups $C$ of $G$ generate $H_1(G,M)$?

I expect the answer "Yes". Note that when $G$ is cyclic, the answer is obviously "Yes".