If f is a surjective, but not injective morphism of groups G->G', can there be a cyclic group of G' whose preimage is not cyclic?

Question

If $f$ is a surjective, but not injective morphism of groups $G\to G'$, can there be a cyclic group of G' whose preimage is not cyclic?

Edit: I am almost sure the answer is yes, but I would like to see an example.

Answer 1

The group homomorphism $F(2) \rightarrow \Bbb Z$ sending both generators of the free group on 2 elements $F(2)$ to 1 is surjective, not injective and the preimage of $\Bbb Z$ is $F(2)$, hence not cyclic.

Answer 2

Yes. Consider $f : S_{42} \rightarrow \lbrace e \rbrace$, where $S_{42}$ denotes the symetric group over $42$ elements, and $\lbrace e \rbrace$ denotes the trivial group, defined by $$f(\sigma)=e$$

The preimage of the cyclic group $\lbrace e \rbrace$ is isomorphic to $S_{42}$ which is not cyclic.