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".